KIDs like cones

Consider the (future) light-cone C_O issued from a point O in an (n + 1)-dimensional spacetime $({\mathscr M},g)$, n ⩾ 3; by this we mean the subset of ${\mathscr M}$ covered by future-directed null geodesics issued from O. (We expect that our results remain true for n = 2; this requires a more careful analysis of some of the equations arising, which we have not attempted to carry out.) Let (x^μ) = (x⁰, xⁱ) = (x⁰, r, x^A) be a coordinate system such that x⁰ vanishes on C_O. In the theorem that follows the initial data for the sought-for Killing vector field are provided by a spacetime vector field $\overline{Y}{}$ which is defined on C_O only.¹ We will need to differentiate $\overline{Y}{}$ in directions tangent to C_O, for this we need a covariant derivative operator which involves only derivatives tangent to the characteristic hypersurfaces. In a coordinate system such that the hypersurface under consideration is given by the equation x⁰ = 0, for the first derivatives the usual spacetime covariant derivative $\nabla _i \overline{Y}{}^\mu$ applies. However, the tensor of second spacetime covariant derivatives involves the undefined fields $\nabla _ 0 \overline{Y}{}^\mu$. To avoid this we set, on the hypersurface {x⁰ = 0},²

$$\begin{equation} {\mathrm{D}}_{i} \overline{Y}{}_ \mu := \partial _ i \overline{Y}{}_\mu - \Gamma ^\nu _{\mu i} \overline{Y}{}_\nu \ \equiv \ \nabla _i \overline{Y}{}_\mu , \end{equation} \tag{ 1.2 }$$

$$\begin{eqnarray} {\mathrm{D}}_{i} {\mathrm{D}}_j \overline{Y}{}_ \mu &:= & \partial _ i {\mathrm{D}}_j \overline{Y}{}_\mu - \Gamma ^k_{ij} {\mathrm{D}}_k \overline{Y}{}_\mu - \Gamma ^\nu _{i\mu } {\mathrm{D}}_j \overline{Y}{}_\nu \nonumber \\ & \equiv & \partial _ i \nabla _j \overline{Y}{}_\mu - \Gamma ^k_{ij} \nabla _k \overline{Y}{}_\mu - \Gamma ^\nu _{i\mu } {\mathrm{D}}_j \overline{Y}{}_\nu , \end{eqnarray} \tag{ 1.3 }$$

with an obvious similar formula for ${\mathrm{D}}_{i} {\mathrm{D}}_j \overline{Y}{}^\mu$. When the restriction $\overline{\nabla _0 Y{}_ \mu }{}$ to the hypersurface {x⁰ = 0} of the x⁰-derivative is defined we have

$$\begin{equation*} {\mathrm{D}}_{i} {\mathrm{D}}_j \overline{Y}{}_\mu = \nabla _i \nabla _j \overline{Y}{}_\mu +\Gamma ^0_{ij}\overline{ \nabla _0 Y_\mu }{} . \end{equation*}$$

Clearly, ${\mathrm{D}}_{i} {\mathrm{D}}_j \overline{Y}{}_0$ coincides with $\nabla _i \nabla _j Y_0 |_{\lbrace x^0=0\rbrace }$ when $\overline{Y}{} ^\mu$ is the restriction to the hypersurface {x⁰ = 0} of a Killing vector field Y^μ, as then ∇₀Y₀ = 0. This is of key importance for our equations below.

In the adapted null coordinates of [2] we have $\Gamma ^0_{1i}=0$ on {x⁰ = 0} (see [2, appendix A]), so in these coordinates D_iD_j differs from ∇_i∇_j only when i, j ∈ {2, ..., n}.

Our main result is:

Theorem 1.1. Let $\overline{Y}{}$ be a continuous vector field defined along C_O in a vacuum spacetime $({\mathscr M},g)$, smooth on C_O∖{O}. There exists a smooth vector field X satisfying the Killing equations on D⁺(C_O) and coinciding with $\overline{Y}{}$ on C_O if and only if on C_O it holds that

$$\begin{equation} {\mathrm{D}}_i \overline{Y}{}_j + {\mathrm{D}}_j \overline{Y}{}_i =0 , \end{equation} \tag{ 1.4 }$$

$$\begin{equation} {\mathrm{D}}_{1} {\mathrm{D}}_1\overline{Y}_0 = R_{011}{}^{\mu }\overline{Y}_{\mu } . \end{equation} \tag{ 1.5 }$$

Furthermore, (1.5) is not needed on the closure of the set on which the divergence τ of C_O is non-zero.

While this is not necessary, the analysis of KIDs on light-cones thus can be split into two cases: the first is concerned with the region sufficiently close to the tip of the cone where the expansion τ has no zeros. Once a spacetime with Killing field has been constructed near the vertex, the initial value problem for the remaining part of the cone can be reduced to a characteristic initial value problem with two transversally intersecting null hypersurfaces, which will be addressed in theorem 1.2. From this point of view the key restriction for light-cones is (1.4).

The proof of theorem 1.1 can be found in section 2.5. In order to prove it we will first establish some intermediate results, theorems 2.1 and 2.5 below, which require further hypotheses. It is somewhat surprising that these additional conditions turn out to be automatically satisfied.

Equations (1.4) provide thus necessary-and-sufficient conditions for the existence, to the nearby future of O, of a Killing vector field. They can be viewed as the light-cone equivalent of the spacelike KID equations, keeping in mind that (1.5) should be added when C_O contains open subsets on which τ vanishes. As made clear by the definitions, (1.4) and (1.5) involve only the derivatives of $\overline{Y}{}$ in directions tangent to C_O.

We shall see in section 2.6 that some of the equations (1.4) can be integrated to determine $\overline{Y}{}$ in terms of data at O. Once this has been done, we are left with the trace-free part of ${{\mathrm{D}}_A \overline{Y}{}_B + {\mathrm{D}}_B \overline{Y}{}_A}=0$ as the 'reduced KID equations'.

It should be kept in mind that a Killing vector field satisfies an overdetermined system of second-order ODEs which can be integrated along geodesics starting from O, see (2.56) below. This provides both X, and its restriction $\overline{X}{}$ to C_O, in a neighborhood of O, given the free data X^α|_O and ∇^[αX^β]|_O. We will see in the course of the proof of theorem 2.1 how such a scheme ties-in with the statement of the theorem, cf in particular section 2.3.

In section 2.6 we give a more explicit form of the KID equations (1.4) on a cone and discuss some special cases.

In section 3.3 we show how bifurcate Killing horizons arise from totally geodesic null surfaces normal to a spacelike submanifold S of co-dimension two, and how isometries of S propagate to the spacetime.

Throughout, we employ the symbol ($\breve{\cdot }$) to denote the trace-free part of the field (·) with respect to $\tilde{g} = \overline{g}{}_{AB}\, \mathrm{d}x^A\mathrm{d}x^B$. Further, an overbar denotes restriction to the initial surface.

In what follows the coordinates x^A are assumed to be constant on the generators of the null hypersurfaces. The analogue of theorem 1.1 for two intersecting hypersurfaces reads:

Theorem 1.2. Consider two smooth null hypersurfaces N₁ = {x¹ = 0} and N₂ = {x² = 0} in an (n + 1)-dimensional vacuum spacetime $({\mathscr M},g)$, with transverse intersection along a smooth (n − 1)-dimensional submanifold S. Let $\overline{Y}{}$ be a continuous vector field defined on N₁∪N₂ such that $\overline{Y}{}|_{N_1}$ and $\overline{Y}{}|_{N_2}$ are smooth. There exists a smooth vector field X satisfying the Killing equations on D⁺(N₁∪N₂) and coinciding with $\overline{Y}{}$ on N₁∪N₂ if and only if on N₁ it holds that

$$\begin{equation} {\mathrm{D}}_2 \overline{Y}_2 =0 , \end{equation} \tag{ 1.6 }$$

$$\begin{equation} {\mathrm{D}}_{(2} \overline{Y}{}_{A)} =0 , \end{equation} \tag{ 1.7 }$$

$$\begin{equation} ({\mathrm{D}}_{(A} \overline{Y}{}_{B)})\breve{} =0 , \end{equation} \tag{ 1.8 }$$

$$\begin{equation} R{}_{122}{}^{\mu } \overline{Y}{}_{\mu }-{\mathrm{D}}_{2}{\mathrm{D}}_2\overline{Y}{}_1 = 0 , \end{equation} \tag{ 1.9 }$$

where D is the analogue on N₁ of the derivative operator (1.2) and (1.3), with identical corresponding conditions on N₂, and on S one needs further to assume that

$$\begin{equation} ({\mathrm{D}}_1 \overline{Y}{}_{2} + {\mathrm{D}}_{2}\overline{Y}{}_1)|_S = 0 , \end{equation} \tag{ 1.10 }$$

$$\begin{equation} g^{AB} {\mathrm{D}}_A \overline{Y}{}_B |_S =0 , \end{equation} \tag{ 1.11 }$$

$$\begin{equation} \partial _i (g^{AB}{\mathrm{D}}_A \overline{Y}{}_B )|_S =0, \quad i=1,2 , \end{equation} \tag{ 1.12 }$$

$$\begin{equation} R_{21A}{}^{\mu } \overline{Y}{}_{\mu } -{\mathrm{D}}_{A}{\mathrm{D}}_{[1} \overline{Y}{}_{2]} |_S = 0 . \end{equation} \tag{ 1.13 }$$

Similarly to theorem 1.1, (1.9) can be replaced by the requirement that $\overline{g}{}^{AB} {\mathrm{D}}_A \overline{Y}{}_B=0$ on regions where the divergence of N₁ is non-zero. An identical statement applies to N₂.

Theorem 1.2 is proved in section 3. As before, (1.6)–(1.13) provide necessary-and-sufficient conditions for the existence, to the future of S, of a Killing vector field. Hence they provide the equivalent of the spacelike KID equations in the current setting. Note that in (1.6)–(1.13) the derivative D coincides with ∇.

We use local coordinates (x⁰ ≡ u, x¹ ≡ r, x^A) adapted to the light-cone as in [2], in the sense that the cone is given by C_O = {x⁰ = 0} . Further, the coordinate x¹ parameterizes the null geodesics emanating to the future from the vertex of the cone, while the x^A's are local coordinates on the level sets {x⁰ = 0, x¹ = const}≅S^{n − 1}, and are constant along the generators. Then the metric takes the following form on C_O:

$$\begin{equation} g|_{C_O} = \overline{g}_{00} ({\,\rm d}x^{0})^{2} + 2 \nu _0 \,{\rm d}x^{0} \,{\rm d}x^{1} + 2 \nu _A\, {\rm d}x^{0}\, {\rm d}x^{A} + \overline{g}_{AB}\, {\rm d}x^{A} \,{\rm d}x^{B} . \end{equation} \tag{ 2.1 }$$

We stress that we do not assume that this form of the metric is preserved under differentiation in the x⁰–direction, i.e. we do not impose any gauge condition off the cone. On C_O the inverse metric reads

$$\begin{equation} g^{\sharp }|_{C_O} = \overline{g}^{11}\partial ^{2}_{1} + 2 \overline{g}^{1A} \partial _r \partial _{A} + 2 \nu ^0 \partial _{0} \partial _r + \overline{g}^{AB}\partial _{A} \partial _{B} , \end{equation} \tag{ 2.2 }$$

with

$$\begin{equation} \nu ^0 = \frac{1}{\nu _0} , \quad \overline{g}^{1A} = -\nu ^0 \overline{g}^{AB} \nu _B , \quad \overline{g}^{11} = (\nu ^0)^{2} (-\overline{g}_{00} + \overline{g}^{AB}\nu _A\nu _B) . \end{equation} \tag{ 2.3 }$$

We start with a weaker version of theorem 2.5 which, moreover, assumes that the vector field $\overline{Y}{}$ there is the restriction to the light-cone of some smooth vector field Y:

Theorem 2.1. Let Y be a smooth vector field defined in a neighborhood of C_O in a vacuum spacetime $({\mathscr M},g)$. There exists a smooth vector field X satisfying the Killing equations on D⁺(C_O) and coinciding with Y on C_O if and only if the equations

$$\begin{equation} {\mathrm{D}}_i \overline{Y}{}_j + {\mathrm{D}}_j \overline{Y}{}_i =0 , \end{equation} \tag{ 2.4 }$$

$$\begin{equation} R_{011}{}^{\mu }\overline{Y}_{\mu }-{\mathrm{D}}_{1}{\mathrm{D}}_1\overline{Y}_0 = 0 , \end{equation} \tag{ 2.5 }$$

$$\begin{equation} R_{01A}{}^{\mu }\overline{Y}_{\mu } -{\mathrm{D}}_{A}{\mathrm{D}}_1 \overline{Y}_0 = 0 , \end{equation} \tag{ 2.6 }$$

$$\begin{equation} g^{AB}( R_{0AB}{}^{\mu }\overline{Y}_{\mu } - {\mathrm{D}}_{A} {\mathrm{D}}_B\overline{Y}_0 ) = 0 , \end{equation} \tag{ 2.7 }$$

are satisfied by the restriction $\overline{Y}{}$ of Y to C_O.

Proof. To prove necessity, let X be a smooth vector field satisfying the Killing equations on D⁺(C_O):

$$\begin{equation} \underbrace{\nabla _\mu X_\nu + \nabla _\nu X_\mu }_{=:A_{\mu \nu }} =0 , \end{equation} \tag{ 2.8 }$$

the tangent components of which give (2.4). It further easily follows from (2.8) that X satisfies

$$\begin{equation} \nabla _{\mu }\nabla _{\nu } X_\sigma = R _{\alpha \mu \nu \sigma } X^\alpha , \end{equation} \tag{ 2.9 }$$

and (2.5)–(2.7) similarly follow; for (2.7) the equation $\overline{A}{}_{00}=0$ is used.

To prove sufficiency, by contracting (2.9) one finds

$$\begin{equation} \Box X^\sigma = - R^\sigma {}_\alpha X^\alpha \end{equation} \tag{ 2.10 }$$

(which equals −λX^σ under (1.1)). So, should a solution X of our problem exist, it will necessarily satisfy the wave equation (2.10).

Now, it follows from e.g. [7, théorème 2] that for any smooth vector field Y defined on ${\mathscr M}$ there exists a smooth vector field X on ${\mathscr M}$ solving (2.10) to the future of O, such that,

$$\begin{equation} \overline{X}^\mu = \overline{Y}^\mu . \end{equation} \tag{ 2.11 }$$

Here, and elsewhere, overlining denotes restriction to C_O. Further $X|_{D^+(C_O)}$ is uniquely defined by $Y|_{C_O}$.

Applying $\Box$ to (2.8) leads to the identity

$$\begin{equation} \Box A_{\mu \nu } \,{=}\, 2\nabla _{(\mu }\Box X_{\nu )} \,{+}\, 4R_{\kappa (\mu \nu )}{}^{\alpha }\nabla ^{\kappa }X_{\alpha } \,{+}\, 2X_{\alpha } \nabla ^{\kappa }R_{\kappa (\mu \nu )}{}^{\alpha } \,{+}\,2R_{\alpha (\mu }\nabla ^{\alpha } X_{\nu )} . \end{equation} \tag{ 2.12 }$$

When X^μ solves (2.10) this can be rewritten as a homogeneous linear wave equation for the tensor field A_μν,

$$\begin{equation} \Box A_{\mu \nu } = -2R_{\mu }{}^{\alpha }{}_{\nu }{}^{\kappa }A_{\alpha \kappa } +2R_{(\mu }{}^{\alpha }A_{\nu )\alpha } -2{\mathscr L}_X R_{\mu \nu } , \end{equation} \tag{ 2.13 }$$

if one notes that, under (1.1) the last term $-2{\mathscr L}_X R_{\mu \nu }$ equals −2λA_μν (and, in fact, cancels with the before-last one, though this cancellation is irrelevant for the current discussion). It follows from uniqueness of solutions of (2.13) that a solution X of (2.10) will satisfy the Killing equation on D⁺(C_O) if and only if

$$\begin{equation} \overline{A}{_{\mu \nu }}=0 . \end{equation} \tag{ 2.14 }$$

But by (2.4) we already have

$$\begin{equation} \overline{A}{_{ij}}=0 , \end{equation} \tag{ 2.15 }$$

so it remains to show that the equations $\overline{A}{_{0\mu }}=0$ hold. (Annoyingly, these equations involve the derivatives $\overline{\partial _0 X_\mu }{}$ which cannot be expressed as local expressions involving only the initial data $\overline{X}{} = \overline{Y}{}$.) The theorem follows now directly from lemma 2.4 below. □

Definition 2.2. It is convenient to introduce, for a given vector field X, the tensor field

$$\begin{equation} S^{(X)}_{\mu \nu \sigma } := \nabla _{\mu }\nabla _{\nu }X_{\sigma } - R^{\alpha }{}_{\mu \nu \sigma }X_{\alpha } . \end{equation} \tag{ 2.16 }$$

Whenever it is clear from the context which vector field is meant we will suppress its appearance and simply write S_μνσ.

Using the algebraic symmetries of the Riemann tensor we find:

Lemma 2.3. It holds that:

• (i)  
  2S_αβγ = 2∇_(αA_β)γ − ∇_γA_αβ,
• (ii)  
  2S_α(βγ) = ∇_αA_βγ,
• (iii)  
  S_[αβ]γ = 0.

Lemma 2.4. Suppose that $\overline{A}{}_{ij}=0$ and $\Box X= -\lambda X$.

• (1)  
  (2.5) is equivalent to $\overline{A}{}_{01}=0 .$
• (2)  
  If (2.5) holds, then (2.6) is equivalent to $\overline{A}{}_{0A}=0 .$
• (3)  
  If (2.5) and (2.6) hold, then (2.7) is equivalent to $\overline{A}{}_{00}=0 .$

Proof. It turns out to be convenient to consider the identity

$$\begin{eqnarray*} &&\nabla ^\mu A_{\mu \nu } \equiv \underbrace{\nabla ^\mu (\nabla _{\mu }X_{\nu }}_{=-R_{\nu \mu } X^\mu } +\nabla _{\nu }X_{\mu } ) =-R_{\nu \mu } X^\mu + \nabla ^\mu \nabla _{\nu }X_{\mu } = \nabla _\nu \nabla ^{\mu }X_{\mu } = \case{1}{2} \nabla _\nu A^\mu {}_\mu . \end{eqnarray*}$$

Thus, it holds that

$$\begin{equation} g^{\alpha \beta }(2 \nabla _\alpha A_{\beta \nu } - \nabla _\nu A _{\alpha \beta }) = 0 . \end{equation} \tag{ 2.17 }$$

In adapted null coordinates (2.17) implies

$$\begin{eqnarray} &&0 = 2\nu ^0 ( \overline{\nabla _0 A_{1\mu }}{}+ \nabla _1 \overline{A}{}_{0 \mu } - \nabla _{\mu } \overline{A}{} _{01}) + 2\overline{g}{}^{1B}( \nabla _1 \overline{A}{}_{B \mu } + \nabla _B \overline{A}{}_{1 \mu } - \nabla _{\mu } \overline{A}{} _{1B}) \nonumber\\ &&{0 =}+ \overline{g}{}^{11}(2 \nabla _1 \overline{A}{}_{1 \mu } - \nabla _{\mu } \overline{A}{} _{11}) + \overline{g}{}^{BC}(2 \nabla _B \overline{A}{}_{C \mu } - \nabla _{\mu } \overline{A}{} _{BC}) . \end{eqnarray} \tag{ 2.18 }$$

Due to lemma 2.3 we have

$$\begin{equation} \nabla _{0}A_{i j } = \nabla _i A_{j 0} + \nabla _j A_{i 0} - 2S_{ij0} . \end{equation} \tag{ 2.19 }$$

When (ij) = (11) that yields

$$\begin{equation} \nabla _0 A_{11} - 2\nabla _1 A_{01} = -2S_{110} . \end{equation} \tag{ 2.20 }$$

Inserting into (2.18) with μ = 1, after some simplifications one obtains

$$\begin{eqnarray*} &&4\nu ^0{\nabla _1 \overline{A}{}_{01}}+2 \overline{g}{}^{1B} \nabla _B\overline{A}{}_{11} - \overline{g}{}^{BC}( \nabla _1 \overline{A}{}_{BC }-2 \nabla _B\overline{A}{}_{C1} )+ \overline{g}{}^{11}\nabla _1\overline{A}{}_{11} = 4 \nu ^0 \overline{S}{}_{110} . \end{eqnarray*}$$

Using the vanishing of the $\overline{\Gamma }{}^0_{i1}$'s [2, appendix A] and the $\overline{A}{}_{ij}$'s, this becomes a linear homogeneous ODE for $\overline{A}{}_{01}$; in the notation of the last reference (where, in particular, τ denotes the divergence of C_O):³

$$\begin{equation} (2 \partial _r + \tau -2\nu ^0\partial _r \nu _0 )\overline{A}{}_{01} = 2\overline{S}{}_{110} . \end{equation} \tag{ 2.21 }$$

If $\overline{A}{}_{01}=0$, the vanishing of $\overline{S}{}_{110}$ immediately follows.

To prove the reverse implication, for definiteness we assume here and in what follows a coordinate system as in [2, section 4.5]. In this coordinate system τ behaves as (n − 1)/r for small r, ν₀ satisfies ν₀ = 1 + O(r²), and (2.21) is a Fuchsian ODE with the property that every solution which is o(r^{−(n − 1)/2}) for small r is identically zero, see appendix A. As $\overline{A}{}_{01}$ is bounded, when $\overline{S}{}_{110}$ vanishes we conclude that

$$\begin{equation} \overline{A}{}_{01}=0 . \end{equation} \tag{ 2.22 }$$

This proves point 1 of the lemma.

Next, (2.18) with μ = D reads

$$\begin{eqnarray} &&0 = 2\nu ^0( \overline{\nabla _0 A_{1 D}}{}+ \nabla _1 \overline{A}{}_{0 D} - \nabla _D \overline{A}{} _{01}) + 2 \overline{g}{}^{1B}( \nabla _1 \overline{A}{}_{B D} + \nabla _B \overline{A}{}_{1D} - \nabla _D \overline{A}{} _{1B}) \nonumber\\ &&{0 =}+ \overline{g}{}^{11}(2 \nabla _1 \overline{A}{}_{1 D} - \nabla _D \overline{A}{} _{11}) + \overline{g}{}^{BC}(2 \nabla _B \overline{A}{}_{C D} - \nabla _D \overline{A}{} _{BC}) . \end{eqnarray} \tag{ 2.23 }$$

Using (2.19) with (ij) = (A1),

$$\begin{equation} \nabla _0A_{1A} = \nabla _AA_{01} + \nabla _1A_{0A} -2S_{A10} , \end{equation} \tag{ 2.24 }$$

to eliminate $\overline{\nabla _0 A_{1 D}}{}$ from (2.23), and invoking (2.22), on C_O one obtains a system of Fuchsian radial ODEs for $\overline{A}{}_{0D}$,

$$\begin{equation} \left(2\partial _r + \frac{n-3}{n-1}\tau +2\kappa -2\nu ^0\partial _r \nu _0\right)\overline{A}_{0B} -2\sigma _B{}^C\overline{A}_{0C} = 2\overline{S}{}_{B10} , \end{equation} \tag{ 2.25 }$$

with zero being the unique solution with the required behavior at r = 0 when $\overline{S}{}_{B10}=0$:

$$\begin{equation} \overline{A}{}_{0B}=0 . \end{equation} \tag{ 2.26 }$$

This proves point 2 of the lemma.

Let us finally turn attention to (2.18) with μ = 0:

$$\begin{eqnarray} &&0 = 2 \nu ^0 \nabla _1 \overline{A}{}_{0 0} + 2\overline{g}{}^{1B}( \nabla _1 \overline{A}{}_{B 0} + \nabla _B \overline{A}{}_{1 0} - \overline{\nabla _0 A _{1B}}{}) \nonumber\\ &&{0 =}+ \overline{g}{}^{11}(2 \nabla _1 \overline{A}{}_{1 0} - \overline{\nabla _0 A _{11}}{}) + \overline{g}{}^{BC}(2 \nabla _B \overline{A}{}_{C 0} -\overline{ \nabla _0 A _{BC}}{}) . \end{eqnarray} \tag{ 2.27 }$$

The transverse derivatives $\overline{\nabla _0 A _{1i}}{}$ can be eliminated using (2.20) and (2.24),

$$\begin{equation} 2\nu ^0\nabla _1\overline{A}{}_{00} + 4\overline{g}{}^{1B}\overline{S}{}_{B10} + 2\overline{g}{}^{11}\overline{S}{}_{110} + \overline{g}{}^{BC}(2\nabla _B\overline{A}{}_{0C} - \overline{\nabla _0A_{BC}}{}) =0 . \phantom{xx} \end{equation} \tag{ 2.28 }$$

The remaining one, $\overline{g}{}^{AB}\overline{\nabla _0 A _{AB}}{}$, fulfils the following equation on C_O, which follows from (2.19),

$$\begin{eqnarray*} \overline{g}{}^{AB} \overline{\nabla _0A_{AB}}{} &=& 2 \overline{g}{}^{AB}\nabla _A \overline{A}{}_{0B} - 2 \overline{g}{}^{AB} \overline{S}{}_{AB0} = 2 \overline{g}{}^{AB}\nabla _A \overline{A}{}_{0B} -2\tilde{S} - \tau \nu ^0 \overline{A}{}_{00} , \end{eqnarray*}$$

where we have set

$$\begin{equation} \tilde{S} := \overline{g}{}^{AB} \overline{S}{}_{AB0} -\case{1}{2}\tau \nu ^0 \overline{A}{}_{00} . \end{equation} \tag{ 2.29 }$$

Note that $\tilde{S}$ is the negative of the left-hand side of (2.7), and we want to show that the vanishing of $\tilde{S}$ is equivalent to that of $\overline{A}{}_{00}$. Equation (2.28) with $\overline{A}_{ij}=0$ and $\overline{A}_{0i}=0$ (i.e. $\overline{S}{}_{i10}=0$) yields

$$\begin{equation} \nu ^0 (2 \partial _r + \tau + 4 \kappa - 4\nu ^0\partial _r \nu _0 ) \overline{A}{}_{00} = -2\tilde{S} . \end{equation} \tag{ 2.30 }$$

For $\tilde{S}=0$ this is again a Fuchsian radial ODE for $\overline{A}{}_{0 0}$, with the only regular solution $\overline{A}{}_{00}=0$, and the lemma is proved. □

Let us explore the nature of (1.4). Making extensive use of [2, appendix A], and of the notation there (thus $\kappa \equiv \overline{\Gamma }{}^1_{11}$, $\xi _A\equiv -2\overline{\Gamma }{}^{1}_{1A}$, while $\chi _{A}{}^B=\overline{\Gamma }{}_{1A}^B$ denotes the null second fundamental form of C_O), we find

$$\begin{equation} \overline{A}_{11} = 2(\partial _r -\kappa )\overline{X}_1 , \end{equation} \tag{ 2.31 }$$

$$\begin{equation} \overline{A}_{1A} = \partial _r \overline{X}_A - 2\chi _A{}^B\overline{X}_{B} + (\partial _A + \xi _A)\overline{X}_{1} , \end{equation} \tag{ 2.32 }$$

$$\begin{equation} \overline{A}_{AB} = 2\tilde{\nabla }_{(A} \overline{X}_{B)} + 2\chi _{AB}\overline{X}{}^1 - \nu ^0(2\tilde{\nabla }_{(A}\nu _{B)}-\overline{\partial _0 g_{AB}})\overline{X}_1 . \end{equation} \tag{ 2.33 }$$

For definiteness, in the discussion that follows we continue to assume a coordinate system as in [2, section 4.5], in particular κ = 0 and

$$\begin{equation} \chi _{A}{}^{B} = \frac{1}{r} \delta _{A}{}^{B} + O(r),\quad \xi _A =O(r^2), \end{equation} \tag{ 2.34 }$$

$$\begin{equation} \tau = \frac{n-1}{r} + O(r),\quad \partial _{r}\left(\tau - \frac{n-1}{r}\right) = O(1),\quad \partial _{A} \tau = O(r) , \end{equation} \tag{ 2.35 }$$

$$\begin{equation} \sigma _{A}{}^{B} = O(r),\quad \partial _{r} \sigma _{A}{}^{B} = O(1),\quad \partial _{C} \sigma _{A}{}^{B} = O(r). \end{equation} \tag{ 2.36 }$$

Under (1.4) the left-hand sides of (2.31)–(2.33) vanish. Hence, we can determine $\overline{X}{}_1$ by integrating (2.31),

$$\begin{equation} \overline{X}{}_1 (r,x^A) = c(x^A) , \end{equation} \tag{ 2.37 }$$

for some function of the angles.

We continue by integrating (2.32). This is a Fuchsian ODE for $\overline{X}{}_B$, the solutions of which are of the form

$$\begin{equation} \overline{X}{}_A =r \mathring{{\mathscr D}}_Ac + f_A(x^B)r^2 +O(r^3) , \end{equation} \tag{ 2.38 }$$

where $\mathring{{\mathscr D}}$ is the covariant derivative operator of the unit round metric s on S^{n − 1} where f_A(x^B) is an integration function.

In a neighborhood of O, where τ does not vanish, the component $\overline{X}{} ^1$ can be algebraically determined from the equation $\overline{A}_A{}^A=0$, leading to

$$\begin{equation} \overline{X}{}^1 =-\frac{1}{n-1}\Delta _s c -\frac{r}{n-1}s^{AB} \mathring{{\mathscr D}}_Af_B + O(r^2) , \end{equation} \tag{ 2.39 }$$

where Δ_s is the Laplace operator of the metric s.

The equation $\breve{\overline{A}}_{AB}=0$, where $\breve{\overline{A}}_{AB}$ denotes the $\tilde{g}$-trace free part of ${\overline{A}}_{AB}$, imposes the relations

$$\begin{equation} (\mathring{{\mathscr D}}_A\mathring{{\mathscr D}}_Bc)\breve{} = 0 , \end{equation} \tag{ 2.40 }$$

$$\begin{equation} (\mathring{{\mathscr D}}_{(A}f_{B)})\breve{} = 0 , \end{equation} \tag{ 2.41 }$$

with ($\breve{\cdot }$) denoting here the trace-free part with respect to the metric s.

We wish, now, to relate the values of c and f_A to the values of the vector field $\overline{X}{}$ at the vertex, under the supplementary assumption that $\overline{X}{}$ is the restriction to C_O of a differentiable vector field defined in spacetime. Following [2], we denote by y^μ normal coordinates centered at O. Given the coordinates y^μ the coordinates x^α can be obtained by setting

$$\begin{equation} x^{0}=r-y^{0}, \quad x^{1}=r, \quad x^{A}=\mu ^{A}\left(\frac{y^{i}}{r}\right) , \end{equation} \tag{ 2.42 }$$

for some functions μ^A, so that the x^A form local coordinates on S^{n − 1}, and

$$\begin{equation} r:=\left\lbrace \sum _{i}(y^{i})^{2}\right \rbrace ^{\frac{1}{2}}. \end{equation} \tag{ 2.43 }$$

We underline the components of tensor fields in the y^α-coordinates, in particular

$$\begin{equation} \underline{ {X}{}}_\alpha = \frac{\partial x^\mu }{\partial y^\alpha }X_ \mu , \quad { {X}} _\alpha = \frac{\partial y^\mu }{\partial x^\alpha }\underline{X}{}_\mu , \quad \underline{ {X}} ^\alpha = \frac{\partial y^\alpha }{\partial x^\mu }X^\mu , \quad { {X}} ^\alpha = \frac{\partial x^\alpha }{\partial y^\mu } \underline{X}^\mu . \end{equation} \tag{ 2.44 }$$

For vector fields such that $\underline{X}{}^\mu$ is continuous, we obtain

$$\begin{equation} \overline{X}{}{}_1(0) = X{}_1(0) = \underline{X}{}{}_0(0) + \sum _i \underline{X}{}{}_i(0) \frac{y^i}{r}= -\underline{X}{}{}^0(0) + \sum _i \underline{X}{}{} ^i(0) \frac{y^i}{r} . \end{equation} \tag{ 2.45 }$$

Thus, for such vector fields, $\overline{X}{}{}_1(0)$ is a linear combination of ℓ = 0 and ℓ = 1 spherical harmonics, and contains the whole information about $\underline{X}{}^\alpha (0)$. We conclude that

$$\begin{equation} c(x^A) =-\underline{X}{}{}^0(0) + \sum _i \underline{X}{}{} ^i(0) \frac{y^i}{r} . \end{equation} \tag{ 2.46 }$$

Equation (2.40) will be satisfied if and only if c is of the form (2.46), which can be seen by noting that (2.46) provides a family of solutions of (2.40) with the maximal possible dimension.

To determine f_A when $\underline{X}{}^\mu$ is differentiable at the origin we Taylor expand $\underline{X}{}$ there,

$$\begin{eqnarray*} &&\underline{X}{}_\mu = \underline{X}{}_\mu (0)+ y^j \underline{\partial _j X_\mu }{} (0)+ y^0 \underline{\partial _0 X_\mu }{}(0)+ O(|y|^2) . \end{eqnarray*}$$

so that

$$\begin{equation} \overline{X}{}_A = \frac{\partial y^i}{\partial x^A} \overline{ \underline{X}{}}{}_i = \frac{\partial y^i}{\partial x^A} \left( \underline{X}{}_i (0)+ y^j \underline{\partial _j X_i}{} (0)+ y^0 \underline{\partial _0 X_i }{}(0)\right)+ O(r^2) , \end{equation} \tag{ 2.47 }$$

which determines f_A in terms of $\underline{\partial _\mu X_i}{} (0)$. Equation (2.41), which is the conformal Killing vector field equation on S^{n − 1}, will be satisfied under the hypotheses of theorem 2.1 if and only if $\underline{\partial _i X_j}{}(0)$ is anti-symmetric.

As a next step toward the proof of theorem 1.1, we drop in theorem 2.1 the assumption of $\overline{Y}{}$ being the restriction of a smooth spacetime vector field:

Theorem 2.5. Let $\overline{Y}{}$ be a vector field defined along C_O in a vacuum spacetime $({\mathscr M},g)$. There exists a smooth vector field X satisfying the Killing equations on D⁺(C_O) and coinciding with $\overline{Y}{}$ on C_O if and only if on C_O it holds that

$$\begin{equation} {\mathrm{D}}_i \overline{Y}{}_j + {\mathrm{D}}_j \overline{Y}{}_i =0 , \end{equation} \tag{ 2.48 }$$

$$\begin{equation} R_{011}{}^{\mu }\overline{Y}_{\mu }-{\mathrm{D}}_{1}{\mathrm{D}}_1\overline{Y}_0 = 0 , \end{equation} \tag{ 2.49 }$$

$$\begin{equation} R_{01A}{}^{\mu }\overline{Y}_{\mu } -{\mathrm{D}}_{A}{\mathrm{D}}_1 \overline{Y}_0 = 0 , \end{equation} \tag{ 2.50 }$$

$$\begin{equation} g^{AB}( R_{0AB}{}^{\mu }\overline{Y}_{\mu } - {\mathrm{D}}_{A} {\mathrm{D}}_B\overline{Y}_0 ) = 0 . \end{equation} \tag{ 2.51 }$$

Proof of theorem 2.5. We wish to apply theorem 2.1. The crucial step is to construct the vector field Y needed there. For further reference we note that (2.51) will not be needed for this construction.

In the argument that follows we shall ignore the distinction between $\overline{X}{}$ and $\overline{Y}{}$ whenever it does not matter.

By hypothesis it holds that

$$\begin{equation} \overline{A}_{ij}=0 , \end{equation} \tag{ 2.52 }$$

$$\begin{equation} \overline{S}{}_{i10} = 0 . \end{equation} \tag{ 2.53 }$$

We define an anti-symmetric tensor $\overline{F}{}_{\mu \nu }$ via

$$\begin{eqnarray*} &&{-\overline{F}{}_{0i } =\ }\overline{F}{}_{ij} := \nabla _{[i}\overline{X}{}_{j]} , \\ &&-\overline{F}{}_{0i } = \overline{F}{}_{i0} := \nabla _i \overline{X}{}_0 . \end{eqnarray*}$$

Then

$$\begin{eqnarray*} &&\overline{F}_{1i} \equiv \case{1}{2}\nabla _1\overline{X} _{i} - \case{1}{2}\overline{\nabla _{i}X_1} \equiv \nabla _1 \overline{X}{_{i}} - \case{1}{2}\overline{A}_{1i}\\ &&{\overline{F}_{1i} }= \nabla _1\overline{X}{_{i}} . \end{eqnarray*}$$

Moreover,

$$\begin{eqnarray*} &&2 \nabla _1 \overline{F}{}_{ij} \equiv \nabla _1\nabla _{i}\overline{X}{}_{j} - \nabla _1\nabla _{j}\overline{X}{}_{i} \\ &&{2 \nabla _1 \overline{F}{}_{ij}}\equiv \nabla _{i}\nabla _1\overline{X}{}_{j} - \nabla _{j}\nabla _1 \overline{X}{}_{i} + \overline{R}_{1ij}{}^{\alpha }\overline{X}{}_{\alpha } +\overline{R}_{j 1i}{}^{\alpha }\overline{X}{}_{\alpha } \\ &&{2 \nabla _1 \overline{F}{}_{ij}}\equiv \nabla _{i}\overline{A}{}_{1j} - \nabla _{j}\overline{A}{}_{1i} - \overline{\nabla _i\nabla _jX_1}{} + \overline{\nabla _j\nabla _iX_1}{} - \overline{R}{}_{ij1}{}^{\alpha }\overline{X}{}_{\alpha } \\ &&{2 \nabla _1 \overline{F}{}_{ij}}\equiv \nabla _{i}\overline{A}{}_{1j} - \nabla _{j}\overline{A}{}_{1i} - 2\overline{R}{}_{ij 1}{}^{\alpha }\overline{X}{}_{\alpha } . \end{eqnarray*}$$

With (2.52) that gives

$$\begin{eqnarray*} &&\overline{\nabla _1 F_{ij}} = - \overline{R}_{ij 1}{}^{\alpha }\overline{X}_{\alpha } . \end{eqnarray*}$$

Further,

$$\begin{eqnarray*} &&\nabla _1 \overline{F}{_{i0}} \equiv \nabla _1\nabla _i\overline{X}{}_0 = \nabla _i\nabla _1\overline{X}{}_0 + \overline{R}{}_{1i0}{}^{\alpha }\overline{X}{}_{\alpha } = \overline{R}{}_{01i}{}^{\mu }\overline{X}{}_{\mu } + \overline{R}{}_{1i0}{}^{\alpha }\overline{X}{}_{\alpha } \\ &&{\nabla _1 \overline{F}{_{i0}}} = -\overline{R}{}_{i01}{}^{\mu }\overline{X}{}_{\mu } . \end{eqnarray*}$$

To sum it up, (2.52) and (2.53) imply that the equations

$$\begin{equation} \nabla _1 \overline{X}{_{\mu }} = \overline{F}_{1\mu } , \end{equation} \tag{ 2.54 }$$

$$\begin{equation} \nabla _1 \overline{ F}{_{\mu \nu }} = \overline{R}{}^{\alpha }{}_{1\mu \nu }\overline{X}_{\alpha } \end{equation} \tag{ 2.55 }$$

hold on C_O,

Let $\mathring{X}^\mu = \overline{X}^\mu |_ O$ and $\mathring{F}_{\mu \nu }= \overline{F}{}_{\mu \nu }|_ O$ be the initial data at O needed for solving those equations. These data can be calculated as follows: (2.40) and (2.41) show that $\mathring{{\mathscr D}}_A c$ and f_A are conformal Killing fields on the standard sphere (S^{n − 1}, s). It follows from [13, proposition 3.2] (a detailed exposition can be found in [12, proposition 2.5.1]) that c is a linear combination of the first two spherical harmonics, so that $\mathring{X}^\mu$ can be read off from c using (2.46). Similarly (2.47) can be used to read off $\mathring{F}_{\mu \nu }$ from f_A.

We conclude that, in coordinates adapted to C_O as in (2.42), under the hypotheses of theorem 2.5 the desired Killing vector X is a solution of the following problem:

$$\begin{equation} \left\lbrace \begin{array}{@{}ll@{}}\nabla _1 \overline{X}{}_\mu = \overline{F}{}_{1\mu } , & {on C_O;} \\ \nabla _1 \overline{F}{}_{\alpha \beta }= R_{ \gamma 1\alpha \beta } \overline{X}{}^\gamma , & \hbox{on $C_O$;} \\ \overline{X}{}^\mu = \mathring{X}^\mu , & \hbox{at $O$;} \\ \overline{F}{}_{\mu \nu }= \mathring{F}_{\mu \nu }, & \hbox{at $O$;} \\ \Box X^\mu = -\lambda X^\mu , & \hbox{on $D^+(C_O)$;}\\ {X^\mu }= \overline{X}{}^{\mu }, & \hbox{on $C_O$.} \end{array} \right. \end{equation} \tag{ 2.56 }$$

Note that the first four equations above determine uniquely the initial data $\overline{X}{^\mu }$ on C_O needed to obtain a unique solution of the wave equation for X^μ.

Now, we claim that there exists a smooth vector field Y^μ defined near O so that $\overline{X}^\mu$ is the restriction of Y^μ to the light-cone. To see this, let $\mathring{\ell }^\mu$ be given and define (x^μ(s), Z^μ(s), F_αβ(s)) as the unique solution of the problem

$$\begin{equation} \left\lbrace \begin{array}{@{}ll@{}}\displaystyle \frac{{\rm d}^2 x^\mu }{{\rm d}s^2} + \Gamma ^\mu _{\alpha \beta } \frac{{\rm d}x^\alpha }{{\rm d}s} \frac{{\rm d}x^\beta }{{\rm d}s} =0, & \hbox{} \\ \displaystyle \frac{{\rm d}Z_\mu }{{\rm d}s} - \Gamma ^\alpha _{\mu \beta } Z_\alpha \frac{{\rm d}x^\beta }{{\rm d}s}= {F_{\alpha \mu }}\frac{{\rm d}x^\alpha }{{\rm d}s}, & \hbox{ } \\ \displaystyle \frac{{\rm d} F_{\alpha \beta }}{{\rm d}s} - \Gamma ^\mu _{\alpha \gamma } \frac{{\rm d}x^\gamma }{{\rm d}s} F_{\mu \beta } - \Gamma ^\mu _{\beta \gamma } \frac{{\rm d}x^\gamma }{{\rm d}s} F_{\mu \alpha } = R_{ \gamma \delta \alpha \beta } Z^\gamma \frac{{\rm d}x^\delta }{{\rm d}s}, & \hbox{ } \\ \displaystyle x^{\mu }(0)= 0 , \quad \frac{{\rm d} x^{\mu }}{{\rm d}s}(0)= \mathring{\ell }^\mu ,& \\ \displaystyle F_{\mu \nu }(0)= \mathring{F}_{\mu \nu }, & \\ \displaystyle {Z^\mu }(0)= \mathring{X}^{\mu }. & \end{array} \right. \end{equation} \tag{ 2.57 }$$

For initial values such that x^μ(1) is defined, set

$$\begin{equation} \displaystyle Y^\mu |_{x^\mu (1)} = Z^\mu |_{s=1}. \end{equation} \tag{ 2.58 }$$

It follows from smooth dependence of solutions of ODEs upon initial data that Y^μ is smooth in all initial variables, in particular in $\mathring{\ell }^\mu$. If the x^μ's are normal coordinates centered at O, then $x^\mu (s=1)=\mathring{\ell }^\mu$, which implies that (2.58) defines a smooth vector field in a neighborhood of O. It then easily follows that the restriction of Y^μ to C_O equals $\overline{X}{}^\mu$, as defined by the first four equations in (2.56).

The hypotheses of theorem 2.1 are now satisfied, and theorem 2.5 is proved. □

To prove theorem 1.1 we will use theorem 2.1, together with some of the ideas of the proof of theorem 2.5. We need to show that (1.4) together with the Einstein equations imply both the existence of a smooth extension Y of $\overline{Y}{}$, and that (2.5)–(2.7) hold.

2.5.1. Properties of S_μνσ

Recall the definition

$$\begin{equation} S_{\mu \nu \sigma } \equiv \nabla _{\mu }\nabla _{\nu }X_{\sigma } - R^{\alpha }{}_{\mu \nu \sigma }X_{\alpha } , \end{equation} \tag{ 2.59 }$$

and lemma 2.3.

In the context of theorem 1.1, only those components of the tensor field S_αβγ which do not involve ∂₀-derivatives of X are a priori known. One easily checks:

Lemma 2.6. The components

$$\begin{equation} \mbox{$\overline{S}{}_{ij\mu }$ with $(ij)\ne (AB)$} \end{equation} \tag{ 2.60 }$$

of the restriction to C_O of S_μνσ can be algebraically $\overline{X}_\sigma\equiv \overline{Y}_\sigma, D_i \overline{X}_\sigma\equiv D_i \overline{Y}_\sigma \hbox{ and }D_i D_j \overline{X}_\sigma \equiv D_i D_j \overline{Y}_\sigma$.

We wish, next, to calculate ∇^αS_αβγ and ∇^γS_αβγ. This requires the knowledge of ∇₀X_μ, of ∇₀∇₀X_μ, and even of ∇₀∇₀∇₀X_μ in some equations. For this, let X be any extension of $\overline{X}{}$ from the light-cone to a punctured neighborhood ${\mathscr O}\setminus \lbrace O\rbrace$ of O, so that the transverse derivatives appearing in the following equations are defined. X is assumed to be smooth on its domain of definition, and we emphasize that we do not make any hypotheses on the behavior of the extension X as the tip O of the light-cone is approached. As will be seen, the transverse derivatives of X on C_O drop out from those final formulae which are relevant for us.

We will make use several times of

$$\begin{equation*} \nabla _\alpha R^\alpha {}_{\beta \gamma \delta } =0 , \end{equation*}$$

which is a standard consequence of the second Bianchi identity when the Ricci tensor is proportional to the metric.

We start with ∇^αS_αβγ. Two commutations of derivatives allow us to rewrite the first term in the divergence of S_αβγ over the first index as

$$\begin{eqnarray} \nonumber \nabla ^\alpha \nabla _\alpha \nabla _\beta X_\gamma & = & \nabla ^\alpha ( \nabla _\beta \nabla _\alpha X_\gamma + R_\gamma {}^\sigma {}_{\alpha \beta } X_ \sigma ) \\ & = & \nabla _\beta \nabla ^\alpha \nabla _\alpha X_\gamma + R {}^{\alpha \sigma }{}_{\alpha \beta } \nabla _ \sigma X_ \gamma + 2R_\gamma {}^\sigma {}_{\alpha \beta } \nabla ^\alpha X_ \sigma \nonumber \\ & = & \nabla _\beta \Box X_\gamma + R {}^{ \sigma }{}_{ \beta } \nabla _ \sigma X_ \gamma + 2R_\gamma {}^\sigma {}_{\alpha \beta } \nabla ^\alpha X_ \sigma . \end{eqnarray} \tag{ 2.61 }$$

Hence, since R_αβ = λg_αβ, and using the first Bianchi identity in the second line

$$\begin{eqnarray} \nonumber \nabla ^\alpha S_{\alpha \beta \gamma } & = & \nabla _\beta \Box X_\gamma + \lambda \nabla _ \beta X_ \gamma + 2R_\gamma {}^\sigma {}_{\alpha \beta } \nabla ^\alpha X_ \sigma - R^\sigma {}_{\alpha \beta \gamma {}} \nabla ^\alpha X_ \sigma \\ & = & \nabla _\beta ( \Box X_\gamma + \lambda X_ \gamma ) + R_\gamma {}^{\sigma \alpha }{}_{\beta } A_{\alpha \sigma } . \end{eqnarray} \tag{ 2.62 }$$

Similarly,

$$\begin{eqnarray} \nonumber \nabla ^\alpha S_{\beta \gamma \alpha } & = & \nabla ^\alpha (\nabla _\beta \nabla _\gamma X_\alpha - R^\sigma {}_{\beta \gamma \alpha } X_\sigma ) \\ \nonumber & = & \nabla _\beta \nabla ^\alpha \nabla _\gamma X_\alpha + R_{\gamma }{}^{\sigma \alpha }{}_{\beta } \nabla _\sigma X_\alpha + R_{\alpha }{}^{\sigma \alpha }{}_{\beta } \nabla _\gamma X_\sigma - R^\sigma {}_{\beta \gamma \alpha } \nabla ^\alpha X_\sigma \\ \nonumber & = & \nabla _\beta ( \nabla _\gamma \nabla ^\alpha X_\alpha + R {}^{\sigma }{}_{\gamma } X_\sigma ) + R_{\gamma }{}^{\sigma \alpha }{}_{\beta } \nabla _\sigma X_\alpha + R {}^{\sigma }{}_{\beta } \nabla _\gamma X_\sigma - R^\sigma {}_{\beta \gamma \alpha } \nabla ^\alpha X_\sigma \nonumber \\ & = & \case{1}{2}\nabla _\beta \nabla _\gamma A^\alpha {}_\alpha + \lambda A_{\beta \gamma } . \end{eqnarray} \tag{ 2.63 }$$

Now, on C_O and in coordinates adapted to the cone

$$\begin{eqnarray} \overline{\nabla ^\alpha S_{\alpha \beta \gamma }}{}&=& \nu ^0 (\overline{\nabla _0 S_ { 1\beta \gamma } }{}+ \nabla _1 \overline{S}{}_ {0 \beta \gamma }) + \overline{g}{}^{1A} (\nabla _1\overline{S}{}_ { A\beta \gamma } + \nabla _A\overline{S}{}_ {1 \beta \gamma }) \nonumber\\ &&+ \overline{g}{}^{11} \nabla _1 \overline{S}{}_ {1 \beta \gamma } + \overline{g}{}^{AB} \nabla _A \overline{S}{}_ {B \beta \gamma } , \end{eqnarray} \tag{ 2.64 }$$

while

$$\begin{eqnarray} \overline{ \nabla ^\alpha S_{ \beta \gamma \alpha }}{} &=& \nu ^0(\overline{\nabla _0 S_ { \beta \gamma 1}}{} + \nabla _1 \overline{S}{}_ { \beta \gamma 0}) + \overline{g}{}^{1B} (\nabla _1 \overline{S}{}_ { \beta \gamma B} + \nabla _B \overline{S}{}_ { \beta \gamma 1}) \nonumber\\ &&+ \overline{g}{}^{11} \nabla _1 \overline{S}{}_ { \beta \gamma 1} + \overline{g}{}^{BC} \nabla _B \overline{S}{}_ { \beta \gamma C} . \end{eqnarray} \tag{ 2.65 }$$

In order to handle undesirable terms such as ∇₀S_βγ1 we write

$$\begin{eqnarray} \nonumber \nabla _0 \nabla _\alpha \nabla _\beta X_\gamma & = & \nabla _\alpha \nabla _0\nabla _\beta X_\gamma + R_{\beta }{}^\sigma {}_{0 \alpha } \nabla _{\sigma } X_\gamma + R_\gamma {}^\sigma {}_{0 \alpha } \nabla _{\beta } X_ \sigma \\ & = & \nabla _\alpha (\nabla _\beta \nabla _0 X_\gamma + R_ {\gamma }{}^{\sigma }{}_{0 \beta } X_ \sigma ) + R_{\beta }{}^\sigma {}_{0 \alpha } \nabla _{\sigma } X_\gamma + R_\gamma {}^\sigma {}_{0 \alpha } \nabla _{\beta } X_ \sigma \nonumber \\ & = & \nabla _\alpha (\nabla _\beta (A_{0\gamma }- \nabla _\gamma X_0) + R_ {\gamma }{}^{\sigma }{}_{0 \beta } X_ \sigma ) \nonumber \\ && + R_{\beta }{}^\sigma {}_{0 \alpha } \nabla _{\sigma } X_\gamma + R_\gamma {}^\sigma {}_{0 \alpha } \nabla _{\beta } X_ \sigma . \end{eqnarray} \tag{ 2.66 }$$

2.5.2. Analysis of condition (2.5)

Lemma 2.7. Assume that $\overline{A}{}_{1i}=0$ and $\breve{\overline{A}{}}_{AB}=0$. Then, in vacuum,

$$\begin{equation} \left(\partial _r + \frac{2}{n-1}\tau -\kappa \right)\partial _r (\overline{g}{}^{AB} \overline{A}{}_{AB}) =2 \tau \nu ^0 \overline{S}{}_{110} . \end{equation} \tag{ 2.67 }$$

Proof. By lemma 2.3 the vanishing of $\overline{A}{}_{1i}$ implies

• (i)  
  $\overline{S}{}_{111} =0$,
• (ii)  
  $0=\overline{S}{}_{11A}$, as well as all permutations thereof.

Consider (2.63) with (βγ) = (11). Setting $a:= \overline{g}{}^{AB}\overline{A}{}_{AB}$ we find

$$\begin{eqnarray} \nonumber \overline{\nabla ^{\alpha }S_{11\alpha }}{} &=& \case{1}{2}\nabla _1\nabla _1\overline{A}{}_{\alpha }{}^{\alpha } + \lambda \overline{A}{}_{11} = \nu ^0\nabla _1\nabla _1\overline{A}{}_{01} + \case{1}{2}\overline{g}{}^{AB} \nabla _1\nabla _1 \overline{A}{}_{AB} \\ &=& \nu ^0\nabla _1\nabla _1\overline{A}{}_{01} + \case{1}{2}(\partial _r -\kappa )\partial _r a . \end{eqnarray} \tag{ 2.68 }$$

Due to lemma 2.3 we have

$$\begin{eqnarray*} 2\chi ^{AB}\overline{S}{}_{1AB} &=& \chi ^{AB}\nabla _1 \overline{A}{}_{AB} = \chi ^{AB}(\partial _r \overline{A}{}_{AB} - 2 \chi _A{}^C\overline{A}{}_{BC}) \\ &=& \frac{1}{n-1}\chi ^{AB}[\partial _r (a\overline{g}{}_{AB}) -2a\chi _{AB}] \\ &=&\frac{1}{n-1}\tau \partial _r a . \end{eqnarray*}$$

Using (2.65) with (βγ) = (11), as well as the last equation, and employing again lemma 2.3 we obtain, on C_O,

$$\begin{eqnarray} \nonumber \overline{\nabla ^{\alpha }S_{11\alpha }}{} &=& \nu ^0(\overline{\nabla _0S_{111}}{} + \nabla _1\overline{S}{}_{110}) + \overline{g}{}^{1B}(\nabla _1\overline{S}{}_{11B} + \nabla _B\overline{S}{}_{111}) \\ \nonumber && + \overline{g}{}^{11}\nabla _1\overline{S}{}_{111} + \overline{g}{}^{BC}\nabla _B\overline{S}{}_{11C} \\ \nonumber &=& \nu ^0(\overline{\nabla _0S_{111}}{} + \nabla _1\overline{S}{}_{110}) + \overline{g}{}^{BC}\nabla _B\overline{S}{}_{11C} \\ \nonumber &=& \nu ^0(\overline{\nabla _0S_{111}}{} + \nabla _1\overline{S}{}_{110}) - 2\chi ^{AB} \overline{S}{}_{1AB} +\tau \nu ^0\overline{S}{}_{110} \\ &=& \nu ^0(\overline{\nabla _0S_{111}}{} + \nabla _1\overline{S}{}_{110})- \frac{1}{n-1}\tau \partial _r a + \tau \nu ^0\overline{S}{}_{110} . \end{eqnarray} \tag{ 2.69 }$$

Using (2.66) we find

$$\begin{eqnarray} \nonumber \overline{\nabla _0 S_{111}}{} &=& \overline{\nabla _0\nabla _1\nabla _1X_1}{}=\nabla _1\nabla _1\overline{A}{}_{01} -\nabla _1 \overline{S}{}_{110} + \overline{R}{}_{011}{}^{\mu }\overline{A}{}_{1\mu } \\ &=& \nabla _1\nabla _1\overline{A}{}_{01} -\nabla _1 \overline{S}{}_{110} . \end{eqnarray} \tag{ 2.70 }$$

Equating (2.68) with (2.69), and using the last equation we end up with (2.67). □

Corollary 2.8. In a region where the divergence τ does not vanish (in particular, near the vertex), $\overline{A}{}_{ij}=0$ implies, in vacuum, $\overline{S}{}_{110}=0$.

2.5.3. Analysis of condition (2.6)

Lemma 2.9. Assume that $\overline{A}{}_{ij}=0$ and $\overline{S}{}_{110}=0$. Then, in vacuum, $\overline{S}{}_{A10}=0$.

Proof. From (2.66) we obtain

$$\begin{eqnarray*} &&\nonumber \nabla _0 S_{A11} = \nabla _0\nabla _A\nabla _1 X_1 = \nabla _A \nabla _1 A_{01}- \nabla _A S_{110} + R_{1}{}^\sigma {}_{0 A} A_{\sigma 1} . \end{eqnarray*}$$

This allows us to rewrite (2.65) with (βγ) = (A1) on C_O as

$$\begin{eqnarray} &&\overline{\nabla ^\alpha S_{ A1 \alpha }}{} = \nu ^0 ( \nabla _A \nabla _1 \overline{A}{}_{01}- \nabla _A\overline{S}{}_{110} + \overline{R}{}_{1}{}^\sigma {}_{0 A} \overline{A}{}_{\sigma 1} + \nabla _1 \overline{S}{}_ {A1 0}) \nonumber\\ &&+ \overline{g}{}^{1B} (\nabla _1 \overline{S}{}_ {A1 B} + \nabla _B\overline{S}{}_ { A1 1}) + \overline{g}{}^{11} \nabla _1\overline{S}{}_ {A1 1} + \overline{g}{}^{BC} \nabla _B \overline{S}{}_ { A1 C} . \phantom{x} \end{eqnarray} \tag{ 2.71 }$$

Combining with (2.63), which reads with (βγ) = (A1)

$$\begin{equation*} \nabla ^{\alpha }S_{A1\alpha } = \case{1}{2}\nabla _A\nabla _1 A^{\alpha }{}_{\alpha } + \lambda A_{1A}, \end{equation*}$$

we obtain on the initial surface

$$\begin{eqnarray} &&- \nu ^0 \nabla _1 \overline{S}{}_ {A1 0} = \nu ^0 ( \nabla _A \nabla _1 \overline{A}{}_{01}- \nabla _A \overline{S}{}_{110} + \overline{R}{}_{1}{}^j{}_{0 A}\overline{A}{}_{j 1}) + \overline{g}{}^{1B} (\nabla _1 \overline{S}{}_ {A1 B} +\nabla _B \overline{S}{}_ { A1 1}) \nonumber\\ &&+\, \overline{g}{}^{11} \nabla _1\overline{S}{}_ {A1 1} + \overline{g}{}^{BC} \nabla _B \overline{S}{}_ { A1 C} -\case{1}{2}\nabla _A \nabla _1 \overline{A}{}^\alpha {}_\alpha - \lambda \overline{A}{}_{1A} . \end{eqnarray} \tag{ 2.72 }$$

Assuming $\overline{A}{}_{ij}=0$, lemma 2.3 shows that $\overline{S}{}_{i11}=0= \overline{S}{}_{1i1}=\overline{S}{}_{11i}= \overline{S}{}_{A1B}$. This allows us to rewrite the right-hand side of (2.72) as

$$\begin{eqnarray*} \mbox{rhs} &=& \nu ^0\nabla _A \nabla _1 \overline{A}{}_{01} - \nu ^0\nabla _A \overline{S}{}_{110} + \overline{g}{}^{BC} \nabla _B \overline{S}{}_ { A1 C} + \overline{g}{}^{1B} (\nabla _1 \overline{S}{}_ {A1 B} + \nabla _B\overline{S}{}_ { A1 1})- \case{1}{2}\nabla _A \nabla _1 \overline{A}{}^\alpha {}_\alpha \\ &=& {-} \nu ^0\nabla _A \overline{S}{}_{110} +\overline{g}{}^{BC} \nabla _B \overline{S}{}_ { A1 C} - \case{1}{2}\overline{g}{}^{BC}\nabla _A \nabla _1\overline{A}{}_{BC} - \overline{g}{}^{1B}(\nabla _B\overline{S}{}_{A11} - \nabla _A\nabla _1\overline{A}{}_{1B}) . \end{eqnarray*}$$

Now, using in addition that $\overline{S}{}_{110}=0$,

$$\begin{eqnarray*} \nabla _B\overline{S}{}_{A11} - \nabla _A\nabla _1\overline{A}{}_{1B} &=& \case{1}{2}\overline{\nabla _A (\nabla _BA_{11} - 2\nabla _1A_{1B})}{} \\ &=& -\nabla _A \overline{S}{}_{11B} = - \nu ^0 \chi _{AB} \overline{S}{}_{110} = 0 . \end{eqnarray*}$$

Hence,

$$\begin{eqnarray*} &&\mbox{rhs} = - \nu ^0\nabla _A\overline{S}{}_{110} + \overline{g}{}^{BC} \nabla _B\overline{S}{}_ { A1 C} - \case{1}{2}\overline{g}{}^{BC}\nabla _A \nabla _1\overline{A}{}_{BC} \\ &&{ \mbox{rhs} }= - \nu ^0\nabla _A \overline{S}{}_{110} + \overline{g}{}^{BC}(\nabla _B\overline{S}{}_{A1C} - \nabla _A\overline{S}{}_{B1C}) \\ &&{ \mbox{rhs} }= - \nu ^0\nabla _A \overline{S}{}_{110} - \overline{g}{}^{BC}(\underbrace{2\chi _{[B}{}^D\overline{S}{}_{A]DC} }_{=0 \mathrm{ \; by \; \rm lemma\, 2.3}}-\nu ^0\chi _{BC}\overline{S}{}_{1A0} + \nu ^0 \chi _{AC}\overline{S}{}_{1B0}) \\ &&{ \mbox{rhs} }= - \nu ^0\nabla _A\overline{S}{}_{110} + \tau \nu ^0 \overline{S}{}_{1A0} - \nu ^0\chi _A{}^B \overline{S}{}_{1B0} \\ &&{ \mbox{rhs} }= \tau \nu ^0 \overline{S}{}_{A10} + \nu ^0\chi _A{}^B \overline{S}{}_{B10} , \end{eqnarray*}$$

and thus, again due to $\overline{S}{}_{110}=0$ and lemma 2.3,

$$\begin{equation} (\partial _r +\tau -\nu ^0\partial _r \nu _0)\overline{S}{}_{A10} = 0 . \end{equation} \tag{ 2.73 }$$

But zero is the only solution of this equation which is o(r^{−(n − 1)}), and to be able to conclude that

$$\begin{equation} \overline{S}{}_{A10} = 0 \end{equation} \tag{ 2.74 }$$

we need to check the behavior of $\overline{S}{}_{A10}$ at the vertex. For definiteness we assume a coordinate system as in [2, section 4.5]. Now, by definition,

$$\begin{eqnarray*} \overline{S}{}_{A10} & = & \overline{\nabla _A \nabla _1 X_0 - R_{\mu A 1 0} X^\mu }{} \\ & = & \overline{\partial _A (\partial _r X_0 - \Gamma _{10}^\mu X_\mu ) - \Gamma _{A1}^\mu \nabla _\mu X_0 - \Gamma _{A0}^\mu \nabla _1 X_\mu - R_{\mu A 1 0} X^\mu }{} . \end{eqnarray*}$$

From (2.37)–(2.39) we find

$$\begin{equation} \overline{X}{}_1, \partial _i \overline{X}{}_1 = O(1) , \quad \overline{X}{}_0, \partial _i \overline{X}{}_0, \partial _A \partial _r \overline{X}{}_0 = O(1) , \end{equation} \tag{ 2.75 }$$

$$\begin{equation} \overline{X}{}_A, \partial _B \overline{X}{}_A =O(r) , \ \partial _r \overline{X}{}_A = O(1) . \end{equation} \tag{ 2.76 }$$

Using the formulae from [2, appendix A] one obtains

$$\begin{eqnarray*} &&\overline{S}{}_{A10} = -\overline{ \underbrace{\Gamma _{A1}^0 }_{=0}\nabla _0 X_0 }{} +O(r^{-1})=O(r^{-1}) , \end{eqnarray*}$$

which implies that (2.74) holds, and lemma 2.9 is proved. □

2.5.4. Proof of theorem 1.1

2.5.5. Analysis of condition (2.7)

A straightforward application of lemma 2.3 yields

Lemma 2.10. Assume that $\overline{A}{}_{i \mu }=0$. Then

• (i)  
  $\overline{S}{}_{ijk}=0$,
• (ii)  
  $\overline{S}{}_{110}=\overline{S}{}_{101} = \overline{S}{}_{011} = 0$,
• (iii)  
  $\overline{S}{}_{A10} =\overline{S}{}_{A01}= \overline{S}{}_{1A0} = \overline{S}{}_{0A1} = \overline{S}{}_{10A} = \overline{S}{}_{01A} = 0$.

Lemma 2.11. Consider a smooth vector field X in a vacuum spacetime $({\mathscr M},g)$ which satisfies $\overline{A}{}_{i \mu }=0$ on C_O and $\Box X + \lambda X=0$. Then

$$\begin{equation} \tilde{S} := \overline{g}{}^{AB}\overline{S}{}_{AB0} - \case{1}{2}\tau \nu ^0 \overline{A}{}_{00}= \overline{g}{}^{AB}({\mathrm{D}}_A{\mathrm{D}}_B \overline{X}{}_0 - \overline{R}{}_{0AB}{}^{\mu }\overline{X}{}_{\mu })=0 . \end{equation} \tag{ 2.77 }$$

Proof. Equation (2.62) yields with $\overline{A}{}_{i \mu }=0$, $\Box X + \lambda X=0$ and in vacuum

$$\begin{eqnarray*} &&\overline{g^{AB} \nabla ^{\alpha }S_{\alpha AB}}{} = -(\nu ^0)^2 \overline{R}{}_{11}\overline{A}{}_{00} = -\lambda (\nu ^0)^2 \underbrace{\overline{g}{}_{11}}_{=0}\overline{A}{}_{00} = 0 . \end{eqnarray*}$$

On the other hand, (2.64) gives with lemma 2.3 and 2.10 on C_O,

$$\begin{eqnarray*} \overline{g}{}^{AB} \overline{\nabla ^{\alpha }S_{\alpha AB}}{} &=& \nu ^0 \overline{g}{}^{AB}(\overline{\nabla _0S_{1AB}}{} + \nabla _1\overline{S}{}_{0AB}) + \overline{g}{}^{1C}\overline{g}{}^{AB}(\nabla _1\overline{S}{}_{CAB} + \nabla _C\overline{S}{}_{1AB}) \\ &&+ \overline{g}{}^{11}\overline{g}{}^{AB}\nabla _1\overline{S}{}_{1AB} + \overline{g}{}^{AB}\overline{g}{}^{CD}\nabla _C\overline{S}{}_{DAB} \\ &=& \nu ^0 \overline{g}{}^{AB}(\overline{\nabla _0S_{1AB}}{} + \nabla _1\overline{S}{}_{0AB}) + \overline{g}{}^{AB}\overline{g}{}^{CD}\nabla _C\overline{S}{}_{DAB} \\ &=& \nu ^0\overline{g}{}^{AB}(\overline{\nabla _0S_{1AB}}{} + \nabla _1\overline{S}{}_{0AB}) +\nu ^0 \overline{g}{}^{AB}(\tau \overline{S}{}_{0AB}+ 2\chi _{A}{}^D \overline{S}{}_{D(0B)}) \\ &=& \nu ^0 \overline{g}{}^{AB}(\overline{\nabla _0S_{1AB}}{} - \nabla _1\overline{S}{}_{AB0} + \nabla _1 \nabla _A\overline{A}{}_{0B}) \\ && -\tau \nu ^0 \overline{g}{}^{AB} \overline{S}{}_{AB0} + (\nu ^0)^2(\tau ^2 + |\chi |^2 )\overline{A}{}_{00} . \end{eqnarray*}$$

Moreover, from (2.66) and lemma 2.3 we deduce that, on C_O,

$$\begin{eqnarray*} \overline{g}{}^{AB}\overline{\nabla _0 S_{1AB}}{} &=&\overline{g}{}^{AB}\overline{\nabla _0\nabla _1\nabla _AX_B}{} \\ &=&\overline{g}{}^{AB}\overline{\nabla _1\nabla _A A_{0B}}{} - \overline{g}{}^{AB}\nabla _1\overline{S}{}_{AB0} + \overline{g}{}^{AB}\overline{R}{}_{01A}{}^{\mu }\overline{A}{}_{B\mu } \\ &=&\overline{g}{}^{AB}\overline{\nabla _1\nabla _AA_{0B}}{} - \overline{g}{}^{AB}\nabla _1\overline{S}{}_{AB0} . \end{eqnarray*}$$

Hence,

$$\begin{eqnarray*} 0 &=& 2 \overline{g}{}^{AB} \nabla _1 \nabla _A\overline{A}{}_{0B} - 2 \overline{g}{}^{AB} \nabla _1\overline{S}{}_{AB0} -\tau \overline{g}{}^{AB} \overline{S}{}_{AB0} + \nu ^0(\tau ^2 + |\chi |^2 )\overline{A}{}_{00} \\ &=& 2 \overline{g}{}^{AB} \nabla _A \nabla _1\overline{A}{}_{0B} -2\nu ^0\overline{R}{}_{11}\overline{A}{}_{00} - 2 \overline{g}{}^{AB} \nabla _1\overline{S}{}_{AB0} -\tau \overline{g}{}^{AB} \overline{S}{}_{AB0} \\ &&+ \nu ^0(\tau ^2 + |\chi |^2 )\overline{A}{}_{00} \\ &=& - 2 \overline{g}{}^{AB} \nabla _1\overline{S}{}_{AB0} -\tau \overline{g}{}^{AB} \overline{S}{}_{AB0} + \nu ^0(2\tau \nabla _1 + \tau ^2 - |\chi |^2 )\overline{A}{}_{00} \\ &=& \tau \nu ^0\left(\partial _r + \case{1}{2}\tau -2\overline{\Gamma }{}^0_{01} \right)\overline{A}{}_{00} -\nu ^0( \partial _r \tau - \kappa \tau + |\chi |^2) \overline{A}{}_{00} - 2 \left(\partial _r + \case{1}{2}\tau -\overline{\Gamma }{}^{0}_{10}\right)\tilde{S} . \end{eqnarray*}$$

Using the vacuum constraint [2] $0 = \lambda \overline{g}{}_{11} = \overline{R}{}_{11} = -\partial _r \tau + \kappa \tau -|\chi |^2$, we obtain

$$\begin{eqnarray*} &&0 = {-}2 \left(\partial _r + \case{1}{2}\tau + \kappa - \nu ^0\partial _r \nu _0 \right)\tilde{S} - \tau \nu ^0\left(\partial _r + \case{1}{2}\tau + 2 \kappa - 2\nu ^0\partial _r \nu _0 \right)\overline{A}{}_{00} . \end{eqnarray*}$$

We employ (2.30),

$$\begin{eqnarray*} &&\tilde{S} = -\nu ^0\left(\partial _r + \case{1}{2}\tau + 2 \kappa - 2\nu ^0\partial _r \nu _0 \right)\overline{A}{}_{00} , \end{eqnarray*}$$

which holds since all the hypotheses of lemma 2.4 are fulfilled, to end up with

$$\begin{eqnarray*} (\partial _r + \tau + \kappa - \nu ^0\partial _r \nu _0 )\tilde{S} &=& 0 . \end{eqnarray*}$$

Regularity at O in coordinates as in [2, section 4.5] gives $\tilde{S} = O(r^{-1})$, which implies that $\tilde{S}=0$ is the only possibility. □

2.6.1. KID equations

Theorem 1.1 shows that a vacuum spacetime emerging as solution of the characteristic initial value problem with data on a light-cone possesses a Killing field if and only if the conformal class $\gamma _{AB}=[\overline{g}{}_{AB}]$ of g_AB, which together with κ describes the free data on the light-cone, is such that, in the region where τ has no zeros, the KIDequations $\overline{A}{}_{ij}=0$ admit a non-trivial solution $\overline{Y}{}$. Written as equations for the vector field $\overline{Y}{}$, they read (we use the formulae from [2, appendix A])

$$\begin{equation} (\partial _r - \kappa + \nu ^0\partial _r \nu _0 ) \overline{Y}{}^0=0 , \end{equation} \tag{ 2.78 }$$

$$\begin{equation} \partial _r \overline{Y}{}^A +( \tilde{\nabla }^A\nu _0 -\partial _r \overline{g}{}^{1A}+ \kappa \nu ^A + \nu _0\xi ^A + \nu _0 \tilde{\nabla }^A) \overline{Y}{}^0 = 0 , \end{equation} \tag{ 2.79 }$$

$$\begin{equation} \tau \overline{Y}{}^1 + \tilde{\nabla }_A\overline{Y}{}^A - \case{1}{2}\nu _0(\zeta +\tau g^{11}+2\overline{g}{}^{1A} \tilde{\nabla }_A )\overline{Y}{}^0 = 0 , \end{equation} \tag{ 2.80 }$$

$$\begin{equation} (\tilde{\nabla }_{(A}\overline{Y}{}_{B)})\breve{} +\sigma _{AB}\overline{Y}{}^1 - \nu _0\big(\overline{g}{}^{11} \sigma _{AB} + \breve{\overline{\Gamma }{}}{}^{1}_{AB} \big) \overline{Y}{}^{0} = 0 , \end{equation} \tag{ 2.81 }$$

where σ_A^B denotes the trace-free part of χ_A^B, $\xi ^A:=\overline{g}{}^{AB}\xi _B$ and

$$\begin{equation} \zeta := 2\overline{g}{}^{AB}\overline{\Gamma }{}^1_{AB} + \tau \overline{g}{}^{11} , \end{equation} \tag{ 2.82 }$$

$$\begin{equation} \xi _A := -2\overline{\Gamma }{}^1_{1A} . \phantom{xx} \end{equation} \tag{ 2.83 }$$

The analysis of these equations is identical to that of their covariant counterpart, already discussed in section 2.3. The first three equations, arising from $\overline{A}{}_{1i}=0$ and $\overline{g}{}^{AB} \overline{A}{}_{AB}=0$ determine a class of candidate fields (depending on the integration functions c(x^A) and f_A(x^B), with $\mathring{{\mathscr D}}_A c$ and f_A being conformal Killing fields on (S^{n − 1}, s). Note that it is crucial for the expansion τ to be non-vanishing in order for $\overline{g}{}^{AB} \overline{A}{}_{AB}=0$ to provide an algebraic equation for $\overline{Y}{}^1$. Regardless of whether τ has zeros or not, we can determine $\overline{Y}{}^1$ by integrating radially (1.5), compare remark 3.2 below.

2.6.2. Killing vector fields tangent to spheres

Let us consider the special case where the spacetime admits a Killing field X with the property that $\overline{X}{}^0 = \overline{X}{}^1 =0$ on C_O. The KID equations for the candidate field $\overline{Y}{}$ (2.78)–(2.81) then reduce to

$$\begin{eqnarray*} \partial _r Y^A &=& 0 , \\ \tilde{\nabla }_{(A} Y_{B)} &=& 0 , \end{eqnarray*}$$

which leads us to the following corollary of theorem 1.1.

Corollary 2.12. Consider initial data $\bar{g}_{AB}(r,x^C)\, {\rm d}x^A\, {\rm d}x^B$ for the vacuum Einstein equations (cf, e.g., [5]) on a light-cone C_O. In the resulting vacuum spacetime there exists a Killing field X with $\overline{X}{}^0 = \overline{X}{}^1 =0$ on C_O defined on a neighborhood of the vertex O if and only if the family of Riemannian manifolds

$$\begin{equation*} (S^{n-1},g_{AB}(r,\cdot )\, {\rm d}x^A \,{\rm d}x^B) \end{equation*}$$

admits an r-independent Killing field f^A = f^A(x^B).

2.6.3. Killing vector fields tangent to the light-cone

Let us now restrict attention to those Killing fields which are tangent to the cone C_O, i.e. we assume

$$\begin{equation} \overline{X}{}^0 \,=\, 0 . \end{equation} \tag{ 2.84 }$$

We start by noting that in the coordinates of (2.42) we have

$$\begin{equation*} \underline{X}_\mu = \omega _{\mu \nu } y^\nu + O(|y|^2) , \end{equation*}$$

for an anti-symmetric matrix ω_μν. Hence, quite generally,

$$\begin{equation} { {X}} ^0 = \frac{\partial x^0}{\partial y^\mu } \underline{X}^\mu = - \underline{X}^0 + \frac{y^i}{r} \underline{X}^i = \omega _{0i}y^i + \frac{y^i}{r} (\omega _{ij}y^j-\omega _{0i}y^0) + O(|y|^2) = O(|y|^2) , \end{equation} \tag{ 2.85 }$$

$$\begin{equation} { {X}} ^1 = \frac{\partial r}{\partial y^\mu } \underline{X}^\mu = \omega _{i0}\frac{y^i}{r} y^0 + O(|y|^2) . \end{equation} \tag{ 2.86 }$$

Thus (2.84) does not impose any restrictions on ω_μν, and we have

$$\begin{equation} \displaystyle \overline{ {X}}{} ^1 = \omega _{i0} {y^i} + O(r^2) . \end{equation} \tag{ 2.87 }$$

Next, under (2.84) the KID equations (2.78)–(2.81) for the candidate field $\overline{Y}{}$ become

$$\begin{equation} \partial _r \overline{Y}{}^A = 0 , \end{equation} \tag{ 2.88 }$$

$$\begin{equation} \tau \overline{Y}{}^1 + \tilde{\nabla }_A\overline{Y}{}^A = 0 , \end{equation} \tag{ 2.89 }$$

$$\begin{equation} (\tilde{\nabla }_{(A}\overline{Y}{}_{B)})\breve{} +\sigma _{AB}\overline{Y}{}^1 = 0 , \end{equation} \tag{ 2.90 }$$

or, equivalently (note that $\partial _r \tilde{\Gamma }^B_{AB}= \partial _A\tau$)

$$\begin{equation} \overline{Y}{}^A = f^A(x^B) , \end{equation} \tag{ 2.91 }$$

$$\begin{equation} \partial _r (\tau \overline{Y}{}^1) + f^A\partial _A\tau = 0 , \end{equation} \tag{ 2.92 }$$

$$\begin{equation} \tilde{\nabla }_{(A} f_{B)} +\chi _{AB}\overline{Y}{}^1 = 0 , \end{equation} \tag{ 2.93 }$$

where we have set $f_A := \overline{g}{}_{AB}f^B$. Equations (2.91)–(2.93) provide thus a relatively simple form of the necessary-and-sufficient conditions for existence of Killing vectors tangent to C_O.

If we choose a gauge where τ = (n − 1)/r (cf e.g. [5]), the last three equations become

$$\begin{equation} \overline{Y}{}^A = f^A(x^B) , \end{equation} \tag{ 2.94 }$$

$$\begin{equation} \overline{Y}{}^1 = -\frac{r}{n-1}\tilde{\nabla }_A f^A = -\frac{r}{n-1}\mathring{{\mathscr D}}_A f^A , \end{equation} \tag{ 2.95 }$$

$$\begin{equation} (\tilde{\nabla }_{(A} f_{B)})\breve{} = \sigma _{AB}\frac{r}{n-1}\mathring{{\mathscr D}}_C f^C . \end{equation} \tag{ 2.96 }$$

Note that there are no non-trivial Killing vectors tangent to all generators of the cone, $\overline{Y}{}^A=0$, as (2.95) gives then $\overline{Y}{}^1=0$. This should be contrasted with a similar question for intersecting null hypersurfaces, see section 3.3.

In analogy with the light-cone case let us first prove an intermediate result.

Theorem 3.1. Consider two smooth null hypersurfaces N₁ = {x¹ = 0} and N₂ = {x² = 0} in an (n + 1)-dimensional vacuum spacetime $({\mathscr M},g)$, with transverse intersection along a smooth submanifold S. Let $\overline{Y}{}$ be a vector field defined on N₁∪N₂. There exists a smooth vector field X satisfying the Killing equations on D⁺(N₁∪N₂) and coinciding with $\overline{Y}{}$ on N₁∪N₂ if and only if on N₁ it holds that

$$\begin{equation} {\mathrm{D}}_2 \overline{Y}_2 =0 , \end{equation} \tag{ 3.1 }$$

$$\begin{equation} {\mathrm{D}}_{(2} \overline{Y}{}_{A)} =0 , \end{equation} \tag{ 3.2 }$$

$$\begin{equation} {\mathrm{D}}_{(A} \overline{Y}{}_{B)} =0 , \end{equation} \tag{ 3.3 }$$

$$\begin{equation} R{}_{122}{}^{\mu } \overline{Y}{}_{\mu }-{\mathrm{D}}_{2}{\mathrm{D}}_2\overline{Y}{}_1 = 0 , \end{equation} \tag{ 3.4 }$$

$$\begin{equation} R{}_{12A}{}^{\mu } \overline{Y}{}_{\mu }-{\mathrm{D}}_{A}{\mathrm{D}}_2\overline{Y}{}_1 = 0 , \end{equation} \tag{ 3.5 }$$

$$\begin{equation} g^{AB} (R{}_{1AB}{}^{\mu } \overline{Y}{}_{\mu }-{\mathrm{D}}_{A}{\mathrm{D}}_B\overline{Y}{}_1 )= 0 , \end{equation} \tag{ 3.6 }$$

where D is the analogue on N₁ of the derivative operator (1.2)–(1.3); similarly on N₂; while on S one needs further to assume that

$$\begin{equation} ({\mathrm{D}}_1 Y_{2} + {\mathrm{D}}_{2}Y_1)|_S = 0 . \end{equation} \tag{ 3.7 }$$

Proof. The proof is essentially identical to the proof of theorem 2.1. The candidate field is constructed as a solution of the wave equation (2.10); the delicate question of regularity of $\overline{Y}{}$ needed at the vertex in the cone case does not arise. Existence of the solution in J⁺(N₁∪N₂) follows from [11].

The main difference is that one cannot invoke regularity at the vertex to deduce the vanishing of, say on N₂, $A_{2\mu }|_{N_2}$ from the equations which correspond to (2.21), (2.25) and (2.30). Instead, one needs further to require (3.7) as well as

$$\begin{eqnarray*} &&(\nabla _2 Y_{A} + \nabla _{A}Y_2)|_S = 0 \quad \mathrm{and} \quad \nabla _{2}Y_2|_S = 0 . \end{eqnarray*}$$

However, the last two conditions follow from (3.1) and (3.2) on N₁. □

We prove now our main result for transversally intersecting null hypersurfaces.

Proof of theorem 1.2. We want to show that (1.6)–(1.13) imply that all the remaining assumptions of theorem 3.1, namely (3.1)–(3.7), are satisfied. The conditions (3.1), (3.2), (3.4) and (3.7) follow trivially.

Lemma 2.7, adapted to the intersecting null hypersurfaces-setting, tells us that g^AB∇_(AY_B) vanishes on N₁∪N₂ due to (1.11) and (1.12). Hence (3.3) is fulfilled.

The analogue of lemma 2.9 for two intersecting null hypersurfaces requires, in addition to (1.13), the vanishing of

$$\begin{equation} {\mathrm{D}}_{A}{\mathrm{D}}_{(1} \overline{Y}{}_{2)} |_S = 0 . \end{equation} \tag{ 3.8 }$$

Both (1.13) and (3.8) together imply vanishing initial data for the analogue of the ODE (2.73) in the current setting. Equation (3.8) follows from (1.7), the corresponding equation on N₂, and (1.10). Thus (3.5) is fulfilled.

A straigthforward adaptation of lemma 2.11 to the current setting shows that, say on N₂, $g^{AB}S_{AB2}-\frac{1}{2}\tau g^{12}A_{22}$ vanishes, supposing that it vanishes on S. Using lemma 2.3 (i) we find

$$\begin{eqnarray*} \left(g^{AB}S_{AB2}-\case{1}{2}\tau g^{12}A_{22}\right)|_S &=& g^{AB}\left(\nabla _{(A}A_{B)2} - \case{1}{2} \nabla _2 A_{AB}\right)-\case{1}{2}\tau g^{12}A_{22} \\ &=& 0 , \end{eqnarray*}$$

because of (1.6)–(1.12). Hence also (3.6) is fulfilled. □

Remark 3.2. While τ has no zeros near the tip of a light-cone, for two transversally intersecting null hypersurfaces the expansion τ may vanish even near the intersection. In that case the trace of (3.3) on, say, N₁ will fail to provide an algebraic equation for $\overline{X}{}^2$. Also, corollary 2.8 cannot be applied to deduce the vanishing of $\overline{S}{}_{221}$, equivalently, the validity of (3.4), in the regions where τ vanishes. Instead one can use the second-order ODE (3.4) to find a candidate for $\overline{X}{}^2$, and then lemma 2.7 guarantees that the trace of the left-hand side of (3.3) vanishes when g^ABA_AB|_S = 0 = ∂₂(g^ABA_AB)|_S.

A key notion to the understanding of the geometry of stationary black holes is that of a bifurcation surface. This is a smooth submanifold S of co-dimension two on which a Killing vector X vanishes, with S forming a transverse intersection of two smooth null hypersurfaces so that X is tangent to the generators of each. In our context this would correspond to a KID which vanishes on S, and is tangent to the null generators of the two characteristic hypersurfaces emanating normally from S. In coordinates adapted to one of the null hypersurfaces, so that the hypersurface is given by the equation x¹ = 0, we have $\overline{X} = \overline{X}{}^ 2 \partial _2$, and $\overline{X}{}^ at = \overline{g}{}_{12} \overline{X}{}^ 2 {\rm d}x^ 1$. Then (2.84) holds, and therefore also (2.88)–(2.90) (which correspond to (3.2) and (3.3)). Equations (2.89)–(2.90) show that this is only possible if τ = σ_AB = 0, which implies that translations along the generators of the light-cone are isometries of the (n − 1)-dimensional metric $\overline{g}{}_{AB} {\rm d}x^ A {\rm d}x^ B$. Equivalently, N₁ and N₂ have vanishing null second fundamental forms, which provides a necessary condition for a bifurcate horizon.

Assuming vacuum (as everywhere else in this work), this condition turns out to be sufficient. Let ζ_A be the torsion one-form of S (see, e.g., [3], or (B.4) below). We can use theorem 2.1 to prove (compare [8, proposition B.1] in dimension 3 + 1 and [9, end of section 2] in higher dimensions):

Theorem 3.3. Within the setup of theorem 2.1, suppose that the null second fundamental forms of the hypersurfaces N_a, a = 1, 2, vanish. Then:

• (i)  
  There exists a Killing vector field X defined on D⁺(N₁∪N₂) which vanishes on S and is null on N₁∪N₂.
• (ii)  
  Furthermore, any Killing vector $\hat{Y} = \hat{Y}^A\partial _A$ of the metric induced by g on S extends to a Killing vector X of g on D⁺(N₁∪N₂) if and only if the $\hat{Y}$-Lie derivative of the torsion one-form of S is exact.

Remarks 3.4. 

• (1)  
  Killing vectors as above would exist to the past of S if the past-directed null hypersurfaces emanating from S also had vanishing second null fundamental forms. However, this does not need to be the case, a vacuum example is provided by suitable Robinson– Trautman spacetimes.
• (2)  
  Concerning point 2. of the theorem, when the $\hat{Y}$-Lie derivative of ζ is merely closed the argument of the proof below provides one-parameter families of Killing vectors defined on domains of dependence $D^+({\mathscr O})$ of simply connected subsets ${\mathscr O}$ of S. It would be of interest to find out whether or not the resulting locally defined Killing vectors can be patched together to a global one when S is not simply connected.

Proof. 

• (1)  
  In coordinates adapted to the null hypersurfaces the condition that a Killing vector X is tangent to the generators is equivalent to

  $$\begin{equation} X^{\mu }|_S=0,\quad X^1|_{N_1}= X^2|_{N_2}=0, \quad X^A|_{N_1\cup N_2}=0 , \end{equation} \tag{ 3.9 }$$

  $$\begin{equation} \Longleftrightarrow X_{\mu }|_S=0,\quad X_2|_{N_1}= X_1|_{N_2}=0, \quad X_A|_{N_1\cup N_2}=0 . \end{equation} \tag{ 3.10 }$$

  For simplicity we assume that the generators of the two null hypersurfaces are affinely parameterized, i.e. $\kappa _{N_1}=\kappa _{N_2}=0$. By hypothesis we have

  $$\begin{equation} \tau _{N_1} = \tau _{N_2} =\sigma ^{N_1}_{AB}= \sigma _{AB}^{N_2} =0 . \end{equation} \tag{ 3.11 }$$

  The KID equations (1.6)–(1.13) for the candidate field $\overline{Y}{}$ reduce to

  $$\begin{equation} \partial _2\partial _2\overline{Y}{}_1 -2\Gamma ^{1}_{12}\partial _2\overline{Y}{}_{1} + \big( \big(\Gamma ^{1}_{12}\big)^2 -\partial _2\Gamma ^{1}_{12}\big)\overline{Y}{}_{1}|_{N_1} =0 , \end{equation} \tag{ 3.12 }$$

  $$\begin{equation} \partial _1\partial _1\overline{Y}{}_2 -2\Gamma ^{2}_{12}\partial _1\overline{Y}{}_{2} + \big( \big(\Gamma ^{2}_{12}\big)^2 -\partial _1\Gamma ^{2}_{12}\big)\overline{Y}{}_{2}|_{N_2} =0 , \end{equation} \tag{ 3.13 }$$

  $$\begin{equation} (\partial _1\overline{Y}{}_{2} + \partial _{2}\overline{Y}{}_1)|_S = 0 , \end{equation} \tag{ 3.14 }$$

  $$\begin{equation} \partial _A (\partial _1 \overline{Y}{}_2 - \partial _2 \overline{Y}{}_1 )|_S = 0 . \end{equation} \tag{ 3.15 }$$

  Since $\overline{Y}{}_{\mu }|_S=0$ we need non-trivial initial data $\partial _2\overline{Y}{}_1|_{S}$ and $\partial _1\overline{Y}{}_2|_{S}$ for the ODEs (3.12) and (3.13) for $\partial _2\overline{Y}{}_1|_{N_1}$ and $\partial _1\overline{Y}{}_2|_{N_2}$, respectively.Using the formulae in [2, appendix A], (3.12)–(3.15) can be rewritten as

  $$\begin{equation} \partial _2\partial _2\overline{Y}{}^2|_{N_1} =0 , \end{equation} \tag{ 3.16 }$$

  $$\begin{equation} \partial _1\partial _1\overline{Y}{}^1|_{N_2} =0 , \end{equation} \tag{ 3.17 }$$

  $$\begin{equation} (\partial _1 \overline{Y}{}^1 + \partial _{2}\overline{Y}{}^2)|_S = 0 , \end{equation} \tag{ 3.18 }$$

  $$\begin{equation} \partial _A(\partial _1 \overline{Y}{}^1- \partial _2 \overline{Y}{}^2)|_S = 0 . \end{equation} \tag{ 3.19 }$$

  Hence there remains the freedom to prescribe a constant c ≠ 0 for $\frac{1}{2}(\partial _1 \overline{Y}{}^1- \partial _2 \overline{Y}{}^2)|_S$. (The constant c reflects the freedom of scaling the Killing vector field by a constant, and is related to the surface gravity of the horizon; we will return to this shortly.) By (3.18) one needs to choose $\partial _1\overline{Y}{}^1|_S=c$ and $\partial _2\overline{Y}{}^2|_S=-c$. Together with $\overline{Y}{}^1|_S=\overline{Y}{}^2|_S=0$ the functions $\overline{Y}{}^2|_{N_1}$ and $\overline{Y}{}^1|_{N_2}$ are then determined by (3.16) and (3.17), and existence of the desired Killing vector follows from theorem 2.1.
• (2)  
  By assumption we have, using obvious notation,

  $$\begin{equation} \chi ^{N_1}_{AB} = \chi ^{N_2}_{AB} =0 \quad \Longleftrightarrow \quad \partial _2 g_{AB}|_{N_1} = \partial _1 g_{AB}|_{N_2} =0 . \end{equation} \tag{ 3.20 }$$

Now, the flow of a spacetime Killing vector field which is tangent to S preserves S. This implies that the bundle of null vectors normal to S is invariant under the flow. Equivalently, the image by the flow of a null geodesic normal to S will be a one-parameter family of null geodesics normal to S. This is possible only if the Killing vector field is tangent to both N₁ and N₂. It thus suffices to consider candidate Killing vector fields $\overline{Y}{}$ which satisfy, in our adapted coordinates,

$$\begin{equation} \overline{Y}{}^1|_{N_1}=\overline{Y}{}^2|_{N_2}=0 \quad \Longleftrightarrow \quad \overline{Y}{}_2|_{N_1}=\overline{Y}{}_1|_{N_2}=0 . \end{equation} \tag{ 3.21 }$$

To continue, we need a simple form of the KID equations (1.6)–(1.13), assuming (3.20) and (3.21), and supposing again that the generators of the two null hypersurfaces are affinely parameterized, i.e. $\kappa _{N_1}=\kappa _{N_2}=0$. Using the notation $\hat{Y} \equiv Y^A|_S\partial _A$ we find:

$$\begin{equation} \partial _{2}\overline{Y}{}^{A}|_{N_1} =0 , \quad \partial _{1} \overline{Y}{}^{A}|_{N_2} =0 , \end{equation} \tag{ 3.22 }$$

$$\begin{equation} ( \tilde{\nabla }_{(A} \overline{Y}{}_{B)})\breve{}\ |_{N_1\cup N_2} =0 , \end{equation} \tag{ 3.23 }$$

$$\begin{equation} \partial _2\partial _2 \overline{Y}{}^{2} |_{N_1}= 0, \quad \partial _1\partial _1\overline{Y}{}^{1} |_{N_2}= 0 , \end{equation} \tag{ 3.24 }$$

$$\begin{equation} (\partial _1 \overline{Y}{}^1 + \partial _{2}\overline{Y}{}^2 +g^{12} {\mathscr L}_{\hat{Y}} g_{12} )|_S = 0 , \end{equation} \tag{ 3.25 }$$

$$\begin{equation} \tilde{\nabla }_A \overline{Y}{}^A |_S =0 , \end{equation} \tag{ 3.26 }$$

$$\begin{equation} [\partial _A (\partial _1\overline{Y}{}^1 - \partial _2\overline{Y}{}^2) + 2{\mathscr L}_{\hat{Y}}\zeta _A]|_S = 0. \end{equation} \tag{ 3.27 }$$

(The fields g₁₂|_S and

$$\begin{equation} \zeta _A = \case{1}{2}(\Gamma ^1_{1A}-\Gamma ^2_{2A})|_S \end{equation} \tag{ 3.28 }$$

are part of the free initial data on S [11]; compare [5].) The first-order equations above are straightforward; some details of the derivation of the remaining equations above will be given shortly. Before passing to that, we observe that (3.22)–(3.23) and (3.26), together with (3.20), are equivalent to the requirement that $\hat{Y}|_S$ is a Killing vector field of (S, g_AB|_S), and that $\overline{Y}{}^A = \hat{Y}^A$ on N₁∪N₂, i.e. that $\overline{Y}{}^A$ is independent of the coordinates x¹ and x². Supposing further that ${\mathscr L}_{ \hat{Y}} \zeta _A$ is exact, the remaining equations (3.24), (3.25) and (3.27) can be used to determine $\overline{Y}{}^1$ and $\overline{Y}{}^2$ on N₁∪N₂. (As such, on each connected component of S the difference $(\partial _1\overline{Y}{}^1 - \partial _2\overline{Y}{}^2)|_S$ is determined up to an additive constant by (3.27), which reflects the freedom of adding a Killing vector field which vanishes on S and is tangent to the null geodesics generating both initial surfaces.) The existence of a Killing vector field X on the spacetime which coincides with $\hat{Y}$ on S follows now from theorem 1.2.

In appendix B it is shown that the KID equations (3.22)–(3.27) are invariant under affine reparameterizations of generators. We also show there that the freedom to choose an affine parameter on N₁ and N₂ can be employed to prescribe g₁₂ on S, and also to add arbitrary gradients to ζ. In particular exactness of ζ, and thereby solvability of the KID equations, is independent of the gauge, as one should expect.

Let us pass to some details of the derivation of the second-order equations above. The calculation uses extensively

$$\begin{equation*} 0=\Gamma ^\mu _{22}|_{N_1} =\Gamma ^A_{B2}|_{N_1} , \end{equation*}$$

similarly on N₂. First, let us compute $R_{122}{}^A|_{N_1}$ and $R_{211}{}^A|_{N_2}$, as needed to evaluate (1.9). Making use of [2, appendix A], we find

$$\begin{eqnarray*} \partial _1\Gamma ^A_{22}|_{N_1} &=& g^{2A}\partial _2\partial _2g_{12} + g^{AB}\partial _2(\partial _1g_{2B} - \partial _Bg_{12}) \\ &=& g^{2A}\partial _2\partial _2g_{12} + \partial _2\big(g_{12}g^{AB}\xi ^{N_1}_B\big) + g^{AB}\partial _2 \partial _2 g_{1B} , \\ \Gamma ^A_{12}|_{N_1} &=& \case{1}{2}g_{12}g^{AB}\xi ^{N_1}_B + g^{AB}\partial _2 g_{1B} + g^{2A}\partial _2 g_{12} , \end{eqnarray*}$$

and thus

$$\begin{eqnarray*} &&R_{122}{}^A|_{N_1} = (\partial _2 +g^{12}\partial _2g_{12} )\Gamma ^A_{12} - \partial _1\Gamma ^A_{22} = \case{1}{2}g_{12}g^{AB} \partial _2\xi ^{N_1}_B , \end{eqnarray*}$$

where, using the notation in [5], $\xi ^{N_1}_A=-2\Gamma ^2_{2A}|_{N_1}$ and $\xi ^{N_2}_A=-2\Gamma ^1_{1A}|_{N_2}$. Now for $\chi ^{N_1}_{AB}=\chi ^{N_2}_{AB}=0$ the vacuum characteristic constraint equations [2] imply $\partial _2\xi ^{N_1}_A=0$ and $\partial _1\xi ^{N_2}_A=0$, hence

$$\begin{equation*} R_{122}{}^A|_{N_1}=R_{211}{}^A|_{N_2}=0 . \end{equation*}$$

Further simple calculations lead to (3.24).

Next, on S, we consider the KID equation

$$\begin{equation} 0= \nabla _A \nabla _1 \overline{Y}{}^1 - R_{\mu A 1}{}^1 \overline{Y}{}^\mu = \nabla _A \nabla _1 \overline{Y}{}^1 + R^1{}_{1 BA} \overline{Y}{}^B . \end{equation} \tag{ 3.29 }$$

Again on S it holds that

$$\begin{eqnarray*} &&\nabla _1 \overline{Y}{}^\mu \partial _\mu = \partial _1 \overline{Y}{}^1 \partial _1 + \Gamma ^\mu _{1 A} \overline{Y}{}^A \partial _\mu , \\ &&\nabla _\mu \overline{Y}{}^1 {\rm d}x^\mu = \partial _1 \overline{Y}{}^1 {\rm d}x^1+ \Gamma ^1_{\mu A}\overline{Y}{}^A {\rm d}x^ \mu , \\ &&\nabla _A \nabla _1\overline{Y}{} ^1 = \partial _A \big(\partial _1 \overline{Y}{}^1 + \Gamma ^1_{1B} \overline{Y}{}^B \big) +\Gamma ^1_{A\mu } \nabla _1 \overline{Y}{}^\mu - \Gamma ^ \mu _{A1} \nabla _\mu \overline{Y}{}^1 \\ &&{\nabla _A \nabla _1\overline{Y}{} ^1 } = \partial _A \partial _1 \overline{Y}{}^1 + \Gamma ^1_{1B} \partial _A \overline{Y}{}^B + \underbrace{\big (\partial _A \Gamma ^1_{1B} + \Gamma ^1_{A\mu }\Gamma ^\mu _{1 B} - \Gamma ^ \mu _{A1} \Gamma ^1_{\mu B}\big )}_{=\partial _B \Gamma ^1_{1A} - R^1{}_{1BA} } \overline{Y}{}^B . \end{eqnarray*}$$

Inserting into (3.29) gives the following equation on S, in coordinates adapted to N₂:

$$\begin{equation} 0 = \nabla _A \nabla _1 \overline{Y}^1 - R_{\mu A 1}{}^1 \overline{Y}^\mu = \partial _A \partial _1 \overline{Y}{}^1 + \Gamma ^1_{1B} \partial _A\overline{Y}{}^B +\partial _B \Gamma ^1_{1A} \overline{Y}{}^B . \end{equation} \tag{ 3.30 }$$

The analogous formula in coordinates adapted to N₁ reads

$$\begin{equation} 0 = \nabla _A \nabla _2 \overline{Y}{}^2 - R_{\mu A 2}{}^2 \overline{Y}{}^\mu = \partial _A \partial _2 \overline{Y}{}^2 + \Gamma ^2_{2B} \partial _A\overline{Y}{}^B +\partial _B \Gamma ^2_{2A} \overline{Y}{}^B . \end{equation} \tag{ 3.31 }$$

Subtracting we obtain (3.27).

From the discussion so far it should be clear that the conditions are necessary. This concludes the proof. □

As an example, suppose that $\hat{Y}$ is a Killing vector on S and that the torsion one-form is invariant under the flow of $\hat{Y}$. It follows from the equations above that we can reparameterize the initial data surfaces so that g₁₂ = 1 on S, with the torsion remaining invariant in the new gauge. Then $\overline{Y}{}^1=\overline{Y}{}^2=0$ and $\overline{Y}{}^A = \hat{Y}^A$ provides a solution of the KID equations on N₁∪N₂.

It is of interest to relate the constant c, arising in the paragraph following (3.19), to the surface gravity (which we denote by $\kappa _{\mathscr H}$ here); this will also prove in which sense the seemingly coordinate-dependent derivatives ∂₁X¹|_S = −∂₂X²|_S are in fact geometric invariants. In the process we recover the well-known fact, that surface gravity is constant on bifurcate horizons. We have

$$\begin{equation} \kappa _{\mathscr H}^2 = - \case{1}{2}(\nabla ^{\mu }X^{\nu })(\nabla _{\mu }X_{\nu })|_{N_1\cup N_2} = -(\nabla _1X^1)(\nabla _2X^2)|_{N_1\cup N_2} . \end{equation} \tag{ 3.32 }$$

On, say, N₁ we have due to (3.9) and (3.16)–(3.19)

$$\begin{equation} \nabla _2X^2|_{N_1} = \partial _2X^2 =- c , \end{equation} \tag{ 3.33 }$$

while $\nabla _1X^1|_{N_1}$ can be computed from (2.10),

$$\begin{eqnarray*} &&\Box X^1|_{N_1} = -R^1{}_{\alpha }X^{\alpha } = -g^{12}R_{22}X^2 =0 \Longleftrightarrow \partial _2\nabla _1X^1|_{N_1} =0 \Longleftrightarrow \nabla _1X^1|_{N_1} =c , \end{eqnarray*}$$

where we used (3.9), the vanishing of χ_AB, and ∇₁X¹|_S = c. Hence

$$\begin{equation} \kappa _{\mathscr H}= |c| . \end{equation} \tag{ 3.34 }$$